That would be up to you. But for me, I would consider it cheating, because you can replicate the sound then look at the keys. It is very difficult for me as well. What I do that helps is to hum the lowest note I hear in the chord, and hold the hum while clicking the chord a few times, and use that to distinguish whether the hummed note is the root, 3rd or 5th.

No such thing as cheating on your musical journey!
It may take away from the pure ear training of the exercises.
One of the big benefits i found since starting to use Tonegym is the application of different chord inversions, intervals, scales etc on my instruments (Guitars & Bass).
This has been very helpful with better mastering the fretboard and opening up new improvisation & songwriting ideas.
For me i'm finding huge benefit in doing this however i do alternate days with / without my instrument on hand too. 😊

It's about improving your ear, use keyboard to identify, then just listen to it a few times before answering the question. Overtime you can use keyboard less, as your ear improves.

I think this might help you. Note that the first point talks about your question, but it's worth reading til the end. I call it "The seven steps (to chord inversions recognition), and I post it here under kind permission of its creator, Yooval Mann.

1) So since this game is advanced I encourage you or I
would recommend using a musical instrument or even an
app on the phone like a virtual piano, don't fret and
don't feel guilt about it, over time you will be able
to drop this aid (promised)

2) use your instrument to detect the lowest note you hear
(the bass)

3) observe the name of the given chord, play the chord
for yourself if needed and observe its structure.

4) determine which degree of the chord does the bass
represents

5) over time when it becomes easier to hear and separate
the bass note, try to detect the next stage's bass by
comparing it to the previous bass you heard, eventually
that would enable you to forego the use of a musical
instrument. But there's no rush.

6) this game requires you to sweat, decide on how much
time you're willing to invest in it and then stick to
the training religiously it doesn't matter if you
finish a level or not but stick to the workout on a
dedicated DAILY religious like routine

7) please report back on how do you do

Thank you all very very much! I will definitely apply your tips and not feel quite so guilty about using my keyboard - but now with a focused method of how to wean myself off of the keyboard.

for 7 chords it helps me to determine where the whole/half step is (between 7th and 1st note of the chord). If there´s no step, just thirds, then it's the root position, if it's at the top then it's the first inversion etc.

I was just wondering something very similar. It's a difficult game. My take on it is this: the aim is to help train our ears -- whether we are "cheating" or not in some sense on the game aspect of it is very much secondary to that. I am simply not at the point where I have any chance of recognizing those inversions just by listening to them without matching them with a keyboard nearby. But that very act of matching, whether it is considered "cheating" or not, is going to help me learn. So the primary goal is achieved.