How thin are you wanting? 2-3oz is plenty thin and can be found on any online leather retailer. Weaver, Tandy, Springfield Leather, Maverick Leather, The Buckle Guy, to name a few.

Sewing machines are a rabbit hole and it depends on what you need. I would check out https://leatherworker.net/forum/forum/50-leather-sewing-machines/

Lots of people asking questions and getting very detailed questions on those forums. Expect to pay for a machine as there are really no leather sewing machines for cheap EXCEPT for the Chinese shoe patcher which is almost it's own project just to get working. I have one and once you figure it out it's not bad and beats hand stitching everything. They have their own Facebook group with tips and advice on getting it just right.

KEKEROSS Manual Leather Sewing Machine Hand Shoe Cobbler Stitching Repair Mending Machine Patcher Dual Cotton Nylon Line Sewing Machine https://www.amazon.com/dp/B08X6KN698/ref=cm_sw_r_apan_glt_fabc_39BGPCZ80NKXJY230SGX?_encoding=UTF8&psc=1