Hi Paul, I am looking into this process as well as our website has not been updated since its original setup in 2008 (!) - it's really just a blog and not very functional. I haven't done a formal RFP, but I've been in touch with firms that I found via Google search and word of mouth, and most of them are happy to meet and provide a proposal. My dream site isn't in the budget right now, so my thinking is to try to find a firm or designer that can provide a scalable, flexible solution, i.e. create something that would serve our needs now and that could be expanded/upgraded into the dream site down the line without requiring a complete overhaul - a design in which all the "bells and whistles", so to speak, could be added on later without having to start from scratch.
Some general guidance would be to determine exactly what you're looking for in the new site and its priorities - obviously the ADA compliance is a priority, but what else? Online collection, virtual exhibitions, etc.? You may want to give the designer a list of 3-5 websites or pages that you like and why - giving them examples is helpful for them to see your vision. With the budget, make sure it includes basic admin costs outside of the design like hosting, domain name registration, maintenance, etc. One proposal I received recommended setting aside another 10-15% of the total budget for additional features or components that you didn't anticipate previously.
Finally, as a good starting point, I'd recommend reaching out to Cuberis. They focus almost exclusively on museum websites and work primarily with Wordpress: https://cuberis.com/. They also have some budgeting tips at their site.
Hope this was helpful! Please feel free to email me if you'd like to hear more about my website redesign adventures so far. :)
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Hillary Belzer
Founder and Curator
The Makeup Museum
Baltimore MD
mmcurator@gmail.com------------------------------