I visited Tulum a year ago and stayed at the Cabanas Copal. It might not quite qualify as “budget”, but it was a truly lovely place to stay, with very friendly staff and a beautiful (clothing optional) beach with a snack bar.
I know there’s other, more rustic places where you can rent a little Cabana for as low as $10 a night, but I found this place a nice compromise between comfort and a wilderness experience. There’s no electricity in the Cabanas, and the rooms have semi-open walls (the walls are built with bamboo-like posts with the space between them not filled in by anything). Luckily, the beds have mosquito nets :). But each cabana has it’s own bathroom, with hot water. I personally preferred it over other places with a bare room with a hammock, and a walk outside to the bathroom.
Be aware that to get to any of the local attractions, you’ll need to get a taxi, rent a bike, or hitch a ride. The Hotel district (on the beach) is about a 10-minute drive from the town. Most of the hotels have their own restaurants, and there’s a few little stores, but most shopping and restaurants that have more variety are in town. There’s also a decent supermarket, and some small produce shops in town, but if you’re staying in an open Cabana, the animals will come in through the gaps in the walls and eat your food!
Make sure you swim in as many Cenotes as you can while you’re in the Yucatan! Swimming in many different bodies of water (the ocean, rivers, and cenotes) was definitely the highlight of my trip. Have fun!