Hi there, Brianna!
Welcome to planDisney! There's no doubt that the best souvenir from a Disneyland Resort visit is the photos we take together. I'm glad you have such a lovely memory of spending the day there with your dad!
The closest thing to a custom portrait at the Disneyland Resort would be a
caricature drawing, however, I'm not sure I'd call that a "Disney-style" cartoon. But there are some seriously talented artists online who offer this type of service if you do a little shopping around! Another option would be to use a filter or generator to upload your photo and create a digital version. A quick web search of "turn my photo into Disney animation" should give you some great ideas!
Once you have the perfect photo you might look for a frame at Disneyland Park!
Plaza Point has some lovely frames, and some of them can even be personalized by the Cast Members in the shop. A frame from Disneyland Park would be an amazing way to display such a special photo!
I'm so glad you have the wonderful memory of your first visit with your dad captured in a photo, Brianna. I hope you'll also have the chance to make more memories at the parks very soon!
Keep moving forward!
Amanda