Welcome to battling!
Yes, the Celestial Tournament is still very do-able, and it's a great experience.
Store pets aren't especially useful there, though. I think the Pandaren Monk can be used in one of the strategies, but I wouldn't expect them to play a major role.
First of all, though, I see your highest level pet is 10, and you need to get at least one pet to 25 before you can even buy more 25s to use in the Celestial Tournament.
The fastest way is through the Draenor Garrison. You can get one instant-25 stone from there. There are other ways, but the Garrison is the fastest. I explain all the methods here:
https://eu.forums.blizzard.com/en/wow/t ... -25/278562
I follow that up with
https://eu.forums.blizzard.com/en/wow/t ... -25/279043
https://eu.forums.blizzard.com/en/wow/t ... ter/286211
https://eu.forums.blizzard.com/en/wow/t ... ris/279486
I actually wrote those for someone who was trying to get through the Celestial Tournament (and she did!)
https://eu.forums.blizzard.com/en/wow/t ... ent/278323
Once you have levelled a pet to 25, you can buy and learn more level 25s.
Look at
https://www.wow-petguide.com/Section/8/ ... Tournament for the set of pets you will need.
My estimate is that you can complete every battle in the game, except the Family achievements, with about 50 pets. The Celestial Tournament will take about 15-20 each week, with A LOT of overlap between the three rortating weeks, so you will want around 30 Blue Level 25 pets in total to complete all of the challenges.
If you are in a tearing hurry, you can buy a lot of the pets at the AH - not all, but a lot - but can get very expensive, so I hope you have some savings.
There is no single team of three pets that is suitable for all battles. If you intend to buy your whole team, and you want to do all three weeks, ummm ... Xu-Fu's is showing that I would need 30 pets to do that with my collection, which is extensive, so figure something like 5K-10K each, with a few actually expensice ones like Sprite Darter Hatchling, so maybe 300K forthe lot, if you can get them already levelled to 25. If you can't, some battler might level them for you if you pay. The going rate used to be about 1K to level a pet 1 to 25. This might be a viable option for you if you have some of the necessary pets in the right breeds already.
Before you do anything, install these addons:
Rematch - lets you import or set up, and remembers your teams
Battle Pet Breed ID - shows you the breeds of pets
Pet Tracker - shows wild pets and shows cooldowns when battling
I honestly think it would be easier, as well as more satisfying, to buy what you can that is cheap, and then work your way up and capture some wild pets. Certainly, choosing and capturing and working towards the Celestial Tournament was the best time I ever had with pets.
Ideally, you would work your way through either the Kalimdor (Horde) or Eastern Kingdom (Alliance) tamers for Taming Kalimdor/EK, then follow the Taming achievements up through Outland, Northrend, Cataclysm and Pandaria, but given that you can now buy 25s, you might opt just to get your shopping list from Xu-Fu's and then capture only the ones you need.
I know that's a lot of information to digest in one lump. Please ask any follow-up questions you may have!