Oh shit I totally forgot about this thread. Well I don't know how well it applies back in the earlier Muromachi, but here's how things would turn out in the Sengoku:

What you have to do depends on your fame and rank really. It's perhaps unsurprising daimyōs themselves seem not to have concerned themselves with this. The ones that rose up the ranks (aka the ones we have records of) seem to have usually came through an introduction of a vassal. How the vassals themselves came by these men likely differed from case to case but it's often the rōnin themselves would make a submission for work.

If you were a daimyō, the vassal would probably vouch for the rōnin. It is not unusually that your vassal would know the rōnin from somewhere (maybe even saw his skills when defeating his lord). But there would indeed be records. One is the the official court record of ranks and clan records of retainer. If it's verifiable that the guy was officially titled guard or protector or something, then he's probably of some ability. Another is the kanjō. A kanjō is basically a decree of praise that is often written by daimyō after a battle or campaign praising a soldier's (or an entire unit's) deeds, and has the issuing daimyō's official seal. A lot were issued, and a lot survive. To get one you have to have really performed exceptionally well and be noticed. Therefore, it was both a huge honour for a soldier to received one, and became an important "letter of recommendation" kind of thing for him when he's looking for work. If the rōnin presents a kanjō, it's sign he's of quite a bit of skill. Given this kind of system, it was not at all unusually for one samurai who don't like the prospects of the clan he's currently serving under to pack his bags up and go look for employment elsewhere.

There were also cases of small bands of mercenaries (kind of like the freelances of Medieval Europe) who would go from lord to lord, battlefield to battlefield looking for work (temporary that hopefully became permanent).

I am actually not sure how exactly people went about looking for samurai not already under their employment. But one way that was used to entice men to come is the equivalent of a signing bonus, giving money to those who show up. Presumably word would be sent out and notice posted that someone was looking for men, and they would come. Money would then be given out based on the equipment and maybe number of men that came, and of course the leader's experience. There were people checking that you actually stayed around to fight after you got your signing bonus, so we know there were people who just got their bonus then left.

If you were actively looking for samurai, there's probably a war to fight. However, before the Edo, the line between samurai and non-samurai was not very clear-cut. Considering the Tanabe family came from Kii but was hired by the Takeda of Kai as gold miners, it's probably not wrong to assume the call for talents would have went out for other purposes too.

Daimyō did try to ensure the loyalty of hires (rōnin or otherwise) by issuing laws against men abandoning posts to serve other masters without the old master's permission, but of course these daimyōs only had authority in their own realm, so such laws were not really effective between realms. With unification though, now such laws could be communicated through the entire country (official Shogunate laws had such mandates too). And with no wars to fight, it became very hard for rōnin to find employment. This lead to poverty and banditry. Rōnins also made up a lot of the anti-Shogunate forces at Osaka and Shimabara. Some gave up their status as samurai to be farmers/merchants. Some even went overseas to be mercenaries (before the ban). Most likely tried to find employment one way or another, as artists, teachers, martial artists.