Hi, welcome, I'll start with my personal favourite: Rios 'Nature', which is half the price of anywhere else with this seller -
Also comes in a lighter brown and black.
For the style of watch, I don't think you can find a better vegan option than the Haveston canvas straps that are heavily covered in this subforum:
I believe ^ these ^ are very much the sort of thing that your style of watch was likely worn on, that or NATO-style nylon webbing.
Or for a large array of non-leather options - take a look at 'Watch-Band-Center' in Germany - plenty of leather options there, too, of course.