The right to vote was restricted in the Antebellum South to substantial property holders, which commonly meant slaveholders. And slavery was integral to the South’s economy, even for non-slaveholders. Similarly, I do not own an oil well, but I want them to keep pumping.
The right to vote was certainly more restricted than it is today but you didn't have to be a "substantial" property holder to vote. Just a property holder in most of the South and even then townies like shop owners, lawyers, etc could vote. The things like literacy tests and poll taxes etc designed to exclude the working class became much more of a feature after the war. Broadly, yeoman farmers could and did vote. You say slavery was integral to the South's economy even for non slave owners. I won't deny all that labor was certainly important as a whole. For non slave owners however, the existence of slavery harmed the value of their labor.