Each year, weeds, which I believe are mostly bermuda grass, take over my garden. I've killed them with sprays (once Roundup but now vinegar); but I understand that this kills only visible growth but not the rhizomes, allowing the same weeds to persist. How can I get rid of the rhizomes and minimize recurring weeds?

Compost and mulch over the weeds. When the emerge again, compost and mulch over them again. Eventually they will smother out.

Bermudagrass is a weed. It is illegal in some states to have seed for it.
The only way to remove it, according to some authorities, is to take a sod cutter or hoe and remove the top 2" of soil. Take it offshore. <g>
Bermudagrass spreads from rhizomes and runners, not by the roots. Therefore, one should never till the soil where bermudagrass is present or it will be a constant nightmare.

Removing and keeping Bermuda grass out of prepared beds is an oft discussed topic. Try some searching, or save the reading time and go to digging instead. Physical removal, physical barriers, weedeaters are your weapons. You can and should mulch your bare soil, but don't count on that to be a single pointed attack...or at least don't expect much success.

As organic1 mentioned, Bermuda grass is sometimes considered a weed, in addition to being a lawn covering and it is aggressive. You have to be more aggressive.