Why are there no pets in Twin Peaks?
From an out of universe point-of-view, it makes complete sense. Pets are not really essential to the story. In-universe, there are clearly enough to keep a reasonably-sized vet open, so we probably just haven't seen them yet.
“Harry, I’m going to let you in on a little secret. Every day, once a day, give yourself a present. Don’t plan it. Don’t wait for it. Just let it happen. It could be a new shirt at the men’s store, a catnap in your office chair, or two cups of good, hot black coffee.” - Dale Cooper
I think you're right that it's a budgetary issue. With this giant cast, something had to give, and not involving a lot of animals and their trainers makes room for more humans to get paid and keeps the production schedule moving along. I'm reminded of the difficult kitten scene in Truffaut's Day for Night.