This generally works
The Immovable Object does not have any specific language calling out it works differently with worn or carried objects, a piece of clothing is definitely an object so it's a valid target for the spell.
Addressing the individual questions:
- Would a creature trapped by an Immovable Object spell on their clothes be considered grappled?
No, nothing in here causes the grappled condition. However, your DM might (and probably should) rule that the creature is Restrained instead. This isn't an effect directly inflicted by the spell, just a natural consequence of what the spell does.
- Would that creature be able to make a check to move the clothes?
Yes, as per the rules for the spell they can make a strength check against the casters spell save DC to move the affected object "up to 10 feet".
- Would that same creature be able to make some kind of acrobatics check to get out of the clothes (no laughter, please, he was wearing a shirt underneath)?
The rules don't say, so the only applicable RAW here is rule 0: The DM decides if that works, if it requires any checks and which if so, if any skill proficiencies apply etc.
Word of advice
RAW this works. RAF this shouldn't work, as a no-save no-attack instant restraint on anything that wears clothes, armor or similar for just a single casting of a 2nd level is broken as all heck. I'd strongly recommend any group that wants to allow Graviturgy wizards to change the spell so that it can't target worn or carried objects.
This applies to a bunch of other features from the Wildemount book as well btw, while a lot of the content from there is pretty cool it's not the most well written material I've ever seen in my life. A lot of the features from that book are unclear or otherwise broken by a strict RAW reading.