Is there gutter around this sunroom?

I'd say it has guttering… but perhaps nowhere to go.

The moulding round the top edge certainly looks gutter-shaped, it has sealed corners, all very 'guttery'.
What it doesn't seem to have is a fall-pipe, so it will either just fill up if those gutter-ends are sealed, or pee down onto the paving if they're not. It also seems designed that the rear face is just going to slosh water against the 'porch' section of the house extension.

It could potentially go right the way round, but the only indication of that is that the interior of the soffit section remains uniform. That, unfortunately doesn't tell us whether or not the outside is uniform. There is potential that it does if you look at the apparently larger gap back to the extension wall in the rear corner. There looks to be room enough for gutter. It would still have no apparent way to drain.

The porch itself seems to have correct guttering & fall-pipe (you can just see a bit of it at the far side & it looks like the same design as the neighbours', it runs to the side of the house. It could potentially be from the conservatory gutter, or the porch, or perhaps both at a push.

I don't see any part of the structure that could have cleverly-hidden integral fall-pipe. None of the uprights are big enough & none of them run down to the ground.

If that was an estate agent's or builder' photo I'd demand to see from more angles. I'd really also like to see one from the top of a ladder looking at that joint to the 'porch'.

One last observation. The door doesn't seem to open onto the paved area. There looks like an accidental half flag of overlap, but mainly it looks like that grass is going to take some footfall, come rain, come shine.

Couple of zoomed images of salient points…