The closest real life example that I know of is RaDVaC. They are professionals in this area, but they are a group of enthusiasts, not a big institution. You can read their white paper to get a basic understanding regarding what was needed to make their vaccine. The rest of the answer is for a peptide vaccine, specifically RaDVaC vaccine, this is the only class of vaccines that I know about in details.
What information is needed for design, and is any non-public?
The most crucial information is the amino acid sequences (peptides) of the virus that are 1) crucial for its functioning, 2) immunogenic, 3) would likely look the same way alone without the neighbours, 4) possible to synthesise. As far as I understood, they got 1 and 2 from public sources (preprints and journal publications), you can follow the references to check if those are actually public. 3 probably requires some special knowledge, but even then you end up with trial and error, so you can probably do without special knowledge. Predicting 4 again requires special knowledge, but there are labs who would synthesise a custom peptide for you for $300 in 4-5 weeks. If they fail to do so, you can probably work with the lab to try some special tricks (amino acid substitution, left/right isomers).
What skills are needed for design (other than general bio/bioinformatics know-how), and how does one design a vaccine?
I am no even sure that you need bio/bioinformatics know-how beyond the stuff that you can read up in the wikipedia.
What equipment and supplies are needed for synthesis? The only complicated ingredient is peptides. You can order their synthesis in a special lab for around $150-$300 per peptide, 4-5 weeks delivery time. The other ingredients you can also buy somewhere. The equipment is trivial, you can buy the whole set for $400 and cook it in your kitchen if you wish.
Does the equipment require any unusual skills (other than typical wetlab stuff, e.g. how to use a pipette or prevent contamination)?
No. Not even wetlab stuff, more like high school chemistry lab skills.
Perhaps most important for a non-professional effort: what quality assurance steps are typically used? How can one verify that the thing one intended to produce was actually produced?
That’s the hard part. Probably not only for a non-professional effort. The simplest answer is you don’t. You just reason that it cannot hurt that much because you know what you have mixed and just try it.
You forgot the most important question: how do you verify that it works? The answer is that you don’t. You definitely cannot do a usual clinical trial with tens of thousands people. The challenge trial doesn’t make much sense if you are only going to use it for yourself and you friend. So probably you just say “ok, it is almost pure safe and there is a chance that it works, so I better make it and take it”. If a lot of people take it, you can have observational data. If people who take are high risk (e.g. doctors) and they don’t get sick compared to their colleagues, then you have some evidence.
There's also a cluster of secondary questions around exactly what one could do with such a vaccine, beyond using it oneself.
You can publish the recipe as RadVac did. Afaik, FDA doesn’t care or cannot do much about it. You can try giving it to other people for a minimal comp. There was another case in US where a guy tried to do so with his own vaccine. FDA sued him and I think it ended up in him having to return money to the “victims”. You can give it to your friends for free, I guess nobody would care (but I don’t live in US). You can sell it in the darknet.
But I've been hearing that the Moderna vaccine was designed in two days, which strongly suggests that it's simple.
The word designed doesn't mean the same as produce. Here it just means deciding for the RNA sequence and that's not a big issue. The sequence for the spike protein is well known and it's basically about chosing a sequence for the spike protein while following a few heuristics about choosing coding that's robust and easily read so that you get a lot of protein production in the cell.
Moderna took 25 days to actually produce their first clinical batch.
Previously they spend 10 years developing know how about how to build the RNA delivery platform.
The fact that CureVac didn't manage to get a vaccine candidate as soon as Moderna/Biontic illustrates that the mRNA vaccine producing step is highly nontrivial.
On the other hand, there are simpler methods to produce vaccines like the one persued by RaDVaC.