The vast majority of trainers are probably a combination of a canvas
upper and a rubber sole: therefore, you do not need to look very
far. However, some brands focus more heavily upon canvas/fabric:
notable converse and vans. The canvas sneaker tends to adhere to
the boating / beach / deck shoe style; the most prominent of which
is the Converse All Star and the Vans Era.
The two shoe's above have an extensive history; both are considered
style icons - sneaker classics - and have been manufactured - unaltered
in design - for over 30 years. Due to the current popularity of
these shoes, many other fashion brands have attempted to cash in
- such as Ralph Lauren. Ultimately, virtually every major sneaker
brand has attempted to produce a canvas deck shoe, so you shouldn't
have much of a problem locating what you are after.