What's a positive word to describe happily stumbling around on a pleasant afternoon, accomplishing nothing in particular?

Only thing that comes to mind is "Saunter," which is a leisurely walk, stroll etcetera.

"Bilbo was enjoying his mead, sauntering around the yard..."

How about the neologism joywalking, the walking version of joyriding? From M-W:

joyride: a ride taken for pleasure (as in a car or aircraft)

Your example:

"Bilbo was enjoying his mead, joywalking around the yard, not accomplishing anything in particular this summer afternoon."

I see no problem with it given that we're talking about Hobbits here. :-)

How about "amble"?

https://www.merriam-webster.com/dictionary/amble

"Bilbo was enjoying his mead, ambling around the yard, not accomplishing anything in particular this summer afternoon."

The origin of "amble" is from a Latin word meaing "walk about", so it would primarily denote a walk. It doesn't depict any parallel activity undertaken while walking.

As to the style and intent of the walk: to walk about suggests that the walk is not undertaken specifically to get from one place to another: "about" in this context feels like walking "to and fro", or perhaps "in a random way".

https://www.vocabulary.com/dictionary/amble

Amble comes from the Latin ambulare, which means "to walk about," as in ambulatory.

Lastly, there is - to me personally - an intangible positive, relaxed quality to the word "amble"; just something about the way it sounds.

That is called pottering (around).

Potter (around): To move around without hurrying, and in a relaxed and pleasant way.

Example: I spent the afternoon pottering around the garden doing a few odd jobs. [Cambridge English dictionary]

---

In your sentence, you could say:

Bilbo was enjoying his mead, pottering around the yard, not accomplishing anything in particular this summer afternoon

Perambulate

to walk about for pleasure

Lollygag

To fool around and waste time.

Meander

to walk slowly without any clear direction:

Stroll may suggest the idea of a pleasant walk:

Walk in a leisurely way.

(Lexico)

"Bilbo was enjoying his mead, strolling around the yard, not accomplishing anything.

pootle, v.

OED British colloquial. intransitive. To move or travel in a leisurely manner; [...] Frequently with around, along, about.

1988 Bicycle Midsummer 57/1 Pootling through Nottingham recently, I came across a branch of a shop called Concept Man.

2020 http://more-to.org/about/ And I love walking, enjoying several expeditions in the greater ranges of the Himalayas and the Andes, as well passionate ramblings around the mountains of Wales. When not walking, you’ll often find me pootling about in the garden, or sitting out with friends and a nice bottle of wine…

No-one so far has mentioned bimble (which is British English):

Bimble at Urban Dictionary:

To amble without real aim, yet in a friendly and harmless manner. It's not required to acheive nothing, though it is a frequent side effect. Bimbling can be made a little more business like with a slight hunch of the shoulders. "Tron and Enid whiled away many a Sunday afternoon on a pleasant bimble round the shops."

Bimble at Wiktionary:

Bimble (noun), (plural bimbles), (chiefly Britain): A gentle, meandering walk with no particular haste or purpose.

Bimble (verb), (third-person singular simple present bimbles, present participle bimbling, simple past and past participle bimbled), (chiefly Britain, intransitive): To walk with no particular haste or purpose.

"Bilbo was enjoying his mead, bimbling around the yard, not accomplishing anything in particular this summer afternoon."

To tootle at lexico.com

‘The lightweight blokart is a micro landsailer, ideal for racing up the beach or for gently tootling along in a light breeze with children.’

‘Just tootling along, getting stuff done, and enjoying.’

‘Sounds particularly great in the car, when tootling round the Peak District on a moody afternoon.’

‘So I tootled over to Minehead, parked, and strolled down to W.H.Smith's, intent on buying a Guardian newspaper.’

‘There you are, two days before the British Grand Prix, minding your own business tootling down the M40 southbound towards Oxford, when suddenly there's a blue flashing light in your rearview mirror.’

‘Goodness knows how many times I've turned the key, waited for the plugs to warm up, started the engine and tootled off merrily without a moment's problem or hesitation.’

‘The Northern Professor and his Godpapa have tootled off down the drive for several days adventure in the north of England.’

The verb laze fulfills both your requirement of moving around lazily while also combining a feeling of contentment. From the OED:

To lie, move, or act in a sleepy listless fashion; to enjoy oneself lazily.

For example:

Bilbo was enjoying his mead, lazing around the yard on this fine summer afternoon.

Bilbo is "loafing" in the back yard.

(from Lexico) idle one's time away, typically by aimless wandering or loitering.

The good thing about loaf is that you can be either moving or stationary, too many of the other answers require you to be moving (which is very un-Hobbit like behavior!).

Not quite single word, but Bilbo could also be "hanging out" in the back yard.

Such a hobbit can be said to be gamboling

Gamboling: to skip about in play : FRISK, FROLIC.

In Middle French, the noun "gambade" referred to the frisky spring of a jumping horse. In the early 1500s, the English word gambol romped into print as both a verb and a noun. (The noun means "a skipping or leaping about in play.") The English word is not restricted to horses, but rather can be used of any frolicsome creature. It is a word that suggests levity and spontaneity, and it tends to be used especially of the lively activity of children or animals engaged in active play.

[Merriam Webster dictionary]

How about messing about? This emphasizes the not doing much aspect while allowing you to not do much while remaining in a single location.

While not English, perhaps we can have a loanword of

wuwei

Which is a kind of 'inaction' that Taoism identifies as have profound good for all of the world.

Besides that, I support the choice of "meander", which has the poetic meaning of streams wandering on their own course and pace, yet being pleasant to all things around them.

"Feelin' groovy"

While you may have a little difficulty transmogrifying the words, this is precisely the state of mind being conveyed by "Feelin' groovy".
Even if this does not end up being what you use (which is likely :-) ) it's worth listening to the song and reading the lyrics to "soak up the ambience"

Video - Simon & Garfunkel - Concert in Central Park - Feellin' Groovy / '59th Bridge Street Song'

Slow down, you move too fast
You got to make the morning last
Just kicking down the cobblestones
Looking for fun and feelin' groovy
Ba da-da da-da da-da, feelin' groovy

Hello lamppost, what'cha knowing
I've come to watch your flowers growin'
Ain't you got no rhymes for me?
Doo-ait-n-doo-doo, feelin' groovy
Ba da-da da-da da-da, feelin' groovy

I got no deeds to do, no promises to keep
I'm dappled and drowsy and ready to sleep
Let the morning time drop all its petals on me
Life, I love you, all is groovy

Legion

Perhaps Bilbo is moseying around the yard?

Mosey (around): 'to walk or go slowly, usually without a special purpose'. [Cambridge Dictionary Online]

So apparently the response above is not detailed enough so here is some further information:

First from the Merriam-Webster dictionary. Note that they provide two definitions

1. to hurry away to move in a leisurely or aimless manner
2. to move in a leisurely or aimless manner: SAUNTER moseyed around the general store — Eric Sevareid

Clearly here I am referring to the second sense, to saunter.

The Oxford English Dictionary online defines it as:

'to walk in a leisurely or aimless manner; to amble, wander.' (Frequently with along, off, on, over).

Interestingly they note that the original meaning was 'to go away quickly or promptly; to make haste' but that this meaning is now rare.

So, perhaps the different usage depends on whether you use British English or American English?

As a British English speaker if, for instance, I was on holidays and I visited a nice tourist spot with my family I might say something like 'let's stop for a coffee and then go for a mosey' and in my postcard home I might say 'we enjoyed moseying around this tourist spot'. It would be understood that I had had a little wander around the town.

I've always liked the word "gallivant"

https://www.merriam-webster.com/dictionary/gallivant

informal : to travel, roam, or move about for pleasure