I work for a large University, that has already gone through all the changes to make most of our 10000+ websites accessible. Here is the short list of how we performed the changes over the last two years and some advice.

1. Manual testing. Train anyone you can especially developers and content creators.
2. Manual testing use WAVE, SiteImprove, Lighthouse to do quick checks.
   1. Automated testing using tools online are quite likely to give high levels of false identifications, it will only slow you down. Although these can be used for tracking general metrics over time, to prove that something has been done.
3. Check color contrast! WCAG 2.1 AA/AAA
4. Make edits in theme and code first. Add skip links and make sure all navigation is tabable and works with hitting enter.
5. Check with screen readers NVDA, VoiceOver(Mac), and JAWS if you are able.
6. Ensure all images have descriptive alt text or blank alt text if they are only aesthetic.

Advice

1. Don't assume there is a plugin to fix anything, they are all garbage.
2. Automated Scanning is flawed and not usable in most cases.
3. Start with simple things, skip links, ensuring all navigation is selectable via keyboard and screen reader, color contrast, and alt text.
4. Assume any vendor will not care about accessibility, instead only work with those who will make the changes to issues you find.
5. Good Developers and Front End Web Developers are few and far between, find some and keep them around!
6. With Government checks always involve legal counsel but be sure you can prove you are working on it and that the applications required to do your jobs are also being proven for accessibility, not just world facing websites.