If the site uses the Block Editor, support is already built in. Add a Group block first, and set its background color. You can then add blocks inside the Group block and set their text colors.
You can define a specific palette for your theme so there are set colors, or you can let WordPress set default color palettes. It also supports custom colors so you can use a color picker to define whatever combinations you want.
It would be wise to educate the client on accessibility as it relates to color contrast. It may also be helpful to educate the client on the ways having defined, predictable color and UI schemes is helpful to visitors who navigate through multiple pages on the site – changing colors frequently may create a jarring or difficult-to-read experience.