You can make a Petsy claim through GapOnly™ which lets you simply pay the difference when you visit the vet, rather than having to wait on being reimbursed. If you don't use this option, your vet might be able to process the claim for you electronically.
Alternatively, you can submit your claim online through the secure Pet Portal. After logging in, you can upload the relevant documents and simply click "Submit". If you prefer, you can also complete a paper claim form and send your original documentation via post.