A word/phrase for "being extremely happy and jumping gladly"

It might take a bit of imagining on the part of the reader, but the following suggestion might evoke the idea of "jumping gladly" that you want:

She was over the moon.

(The link is not to a well-established dictionary, I'm afraid.)

If you really want to describe someone who is actually jumping up and down gladly, you can say:

He/she was literally jumping for joy..

(To improve on Josh's answer)

The expression means:

Be ecstatically happy:

and can be used in situations where one is happy but standing still. According to the definition, physically jumping isn't implied. That's why this is one place where literally could come in handy to mean literally, not figuratively.

[Oxford]

He/she was ... ecstatically jubilant, uncontrollably elated, rambunctious and overjoyed.

You are looking for the word "Ecstasy" and can be described as below.

If you’ve ever been so happy that the rest of the world seemed to disappear, you’ve felt ecstasy — a feeling or state of intensely beautiful bliss.

In your sample question.He was screaming in ecstasy.

Read more about this www.vocabulary.com.

A word that fits the bill pretty perfectly is giddy. This word is used to describe childishly obnoxious joy. A giddy person is pretty much always smiling, usually giggling or laughing, and typically dancing and/or jumping, bouncing, or otherwise being energetic.

He was “skipping with unbridled exuberance”

You could use the word "ebullient" for the cheerful part.

Depending on which dictionary you use, the origin seems to come from 'boiling up' -- a bit of a stretch to equate it with jumping, but it might work for what you need it for.

Exult has the denotation of joy:

verb

[NO OBJECT]
Show or feel triumphant elation or jubilation:

ODO

It contains the connotation of leap:

1560s, "to leap up;" 1590s, "to rejoice, triumph,"
from Middle French exulter,
from Latin exultare/exsultare "rejoice exceedingly, revel, vaunt, boast;"
literally "leap about, leap up,"
frequentative of exsilire "to leap up,"
from ex- "out" (see ex-) + salire "to leap" (see salient (adj.)).
The notion is of leaping or dancing for joy. Related: Exulted; exulting.

She was exulting in her victory

The best single-word answer would be gamboling.

to skip about, as in dancing or playing; frolic.

FULL OF PISS AND VINEGAR... To say that people are “full of piss and vinegar” is to say that they are brimming with energy. Although many speakers assume the phrase must have a negative connotation, this expression is more often used as a compliment, “vinegar” being an old slang term for enthusiastic energy.

Phrasefinder.org: 'Full of piss and vinegar'

Rowdy, boisterous, full of youthful energy.

The earliest citation I've found is from 1936 in John Steinbeck's novel In Dubious Battle:

"Listen, mister," London said, "them guys is so full of piss and vinegar they'll skin you if you show that slick suit outside."

Steinbeck was clearly pleased with the phrase (although it is unlikely that he coined it himself) and repeated the use of it 1938 in his better known novel The Grapes of Wrath:

Grampa walked up and slapped Tom on the chest, and his eyes grinned with affection and pride. "How are ya, Tommy?"

"O.K.," said Tom. "How ya keepin' yaself?"

"Full a piss an' vinegar," said Grampa.