Switch Spaces via Mouse
If you use a Magic Mouse, then yes!
Head over to System Preferences > Mouse. You will find an option there to set it to a two finger swipe to the left/right.

For other mice, ususally no. Some mice, like Logitech Master support all kinds of stuff though with the Logitech Options.
A workaround would be to set a Hot Corner (System Preferences > Mission Control > Hot Corners), then you don't need your keyboard, but it'll take longer.