Tamborim sticks consist of a few plastic rods and a handle holding them together at one end. So keep your eyes open for round plastic objects with a diameter of ideally 5mm and a length of at least 20cm. The bottom part of some coat hangers can be used for that. Whatever you use, make sure the material is at least a litte flexible. If you have some money, you can get plastic rods on the internet. We made good experiences with Polyacetal (POM).
When you have got your material, take about 3 to 7 rods and cut them at the same length (17-25cm) all depending on the flexibility and thickness of the material. Try beating a tambourine to test your decision. Then compound the rods on one side (tape, glue) and stick it into a short piece of a tube, a grip from the handle bar of a bicyle or anything suitable there is. Just make sure everything is tightly fixed.
At the end, grind away pointy spots at the end of the plastic rods to not ruin the tambourine’s skin while playing.