Your Animal Companion makes the check
Your character does not (and, in fact can not) make a Tumble Through check.
Additionally, the only move action you can use is the Mount action to dismount.
Only the Mount would make a Tumble Through check. This makes some sense because they're the one trying to get through the enemy's space, and your character is literally 'along for the ride'; your character's movement is incidental and there is no real mechanic for you to 'fail' to Tumble Through with their mount.
Notably, if a creature has the Tumble Through skill action (see below), there is no further restriction on them using it while with rider.
Alibis: None of Tumble Through, Animal Companions, Animal Companion (list), Minion trait, Mounted Combat, nor Command an Animal Nature skill use provide further guidance on movement.
Your Animal Companion may not be able to Tumble Through (at all)
From the Command an Animal (Nature) action,
Most animals know the Drop Prone, Leap, Seek, Stand, Stride, and Strike basic actions. [...]
Animal Companions alter this in this way
... it gains 2 actions during your turn if you use the Command an Animal action to command it; this is in place of the usual effects of Command an Animal, and you don’t need to attempt a Nature check.
Thankfully (?) @Yopi Lopi pointed out that Tumble Through is a Skill Action; however, it being a Skill Action just makes it even more ambiguous
Anyone can use a skill’s untrained actions, but you can use trained actions only if you have a proficiency rank of trained or better in that skill.
This text is very similar to the Basic Actions, but Command an Animal makes no mention of Skill Actions. RAW, they are able to take the Action but someone attempting to read into it may rule by RAInterpreted against it.
All this is to say, you should bring it up with your GM* if you expect to be doing this frequently. If they believe that Tumble Through is too complex of an action for an animal to attempt naturally, ask if you can use/pay for someone to use the Train Animal Nature Feat to learn it despite it not being a Basic Action.
*I wouldn't consider it very fun to enforce for an Animal Companion anyway, but figured I should include it.