When I read the title, “patience and determination” was the comment I was about to post, but looks like you’ve beat me to it. 
For the Art Academy Mudkip specific, it’s indeed going to be a tough one. With all Art Academy cards, the winners received 100 copies, and there are some extra copies floating around. There are 22 Art Academy cards in total: 10 Japanese; 9 English; 1 German; 1 French; 1 Spanish.
I know some winners have sold more than halve of their 100 cards already, I also know some winners who still have all 100 of their cards and haven’t sold a single one yet. Some pop up sometimes, some haven’t even popped up for sale once.
As for the Mudkip, it’s unfortunately one of the tougher ones to find. I know of two pretty big Art Academy collectors in the western market. One who has 18 out of 22 and one who has 20 out of 22. The one with 18 out of 22 is still missing the Mudkip himself, which should give a pretty good indication just how rare the Mudkip AA card is. The other one with 20 does have the Mudkip however, and there was also a collector at Worlds 2016 whom had a full set of the Japanese AA cards as well (as mentioned in this Efour article), so there is still some hope to one day find this card.
In general however, the best thing to do is:
- Search for them on a regular basis. Not just for sale, but if another collector happens to find it, you might be able to ask them for information if they are willing to share.
- Let as many people as possible know you are willing to buy this card. The more ears out there, the better your chances on finding it. (Especially in combination with a high buying price, and possibly finders fee.)
- Have (sufficient) funds available when you do eventually find it for sale. As a Pikachu collector going after the twelve Pikachu Art Academy cards, this is the toughest one for me. I’m far from rich, and not only have countless of other Pikachu cards I’m looking for, but I also have to save money for my future of course and can’t just throw 10k USD at a single card like it’s nothing. I’m already very happy I have 7 out of the 12 Art Academy Pikachu cards in my collection right now, but I’m pretty sure it will take quite a while before I’ll be able to finish the set of 12 (especially since two are also still missing in both those big Art Academy collector’s collections, including my favorite: the Pikachu Emotions one by Ginga).
Just like with all extremely rare cards, you just have to keep your eyes open, and be ready to strike when the opportunity arises. Although that’s definitely easier said than done, and mostly requires a lot of patience and perseverance, as already mentioned at the top by @papafrankgod .
Greetz,
Quuador