Let me trim this down to a question for which an accurate answer can be given without writing a tome. Let's focus on craft guilds rather than trade guilds (so, carpenters, masons, plasters, etc., rather than merchant guilds). Let's talk, very broadly, about Western Europe and England, and let's use 1500 plus or minus as our date. Based on these, there are a few general rules we can discern. BUT, let's start with a huge caveat: craft guilds in Western Europe in this time were largely wholly locally run--there was usually not an overarching national guild who monitored the activities of local guilds and normalized/standardized operations. So what I'll say will be true for most guilds in most locations most of the time, but a careful scholar will always find exceptions to the stuff I'm going to say, because there was usually little coordination between local guilds. So the rules in one place ain't necessarily the rules in another place. Or from guild to guild inside a single city.
We see a lot of hereditary membership (sons joining their father's guild). The written records left to us suggest that "legacy" guild apprentices were not treated any differently than any other person off the street, which (in my personal opinion) mostly just tells us that organizations have been lying in written records for centuries. But if we just believe what we read, one common way to join a guild would be to have your dad, a guild member, walk you down to the hall to start your initiation/apprenticeship. Typically, an apprentice would serve under a master for the term of his apprenticeship (and, forgive the gendered pronouns, but the sad reality is, these apprentices, journeymen, and masters were, at this time, ALL men), and I know that, in England at least, son could and commonly did apprentice under their fathers when their fathers were masters (although, as someone who has worked for his father...that sounds like a horrible choice).
We also know that guilds accepted truly outside applicants--that is, people unknown to the members of the guild, just some dude off the street. Although a bit before the time period I'm focusing on, in England, urban guilds offered such high wages that rural peasants abandoned the farms they were tending (not owning, just tending) in such high numbers that it (among other causes) induced the 1349 Ordinance of Labourers, which (among other things) prohibited an employer from hiring another employer's employees, a prohibition that was specifically intended to stem the flow of farmers from the countryside to the city. SO, if you got lucky, you could just walk in off the street and join.
A core purpose of the guild is to stabilize costs and revenues, which really means to ensure that the former stay as low as possible and the latter stays as high as possible. Given this, if you happened to show up at the Masons Guild Hall in Bonn during a time when there wasn't a lot of work, you're probably not getting into the guild, and you're certainly not getting into the guild unless you know someone important. Now, that someone need not necessarily be a guild member--if you were the Mayor's brother in law, that may have been enough. Directly to your question, if the guild isn't hiring a lot of people, and you're one of three apprentice candidates, with the other two being (a) the son of one of the masters of the guild and (b) the mayor's brother in law...you better have incriminating engravings showing some of the masters in compromising positions. Because otherwise, you're not getting in.
TL;DR: at the right time and in the right city (which would be specifically a boom time in that particular city), the guilds would have open door hiring, and anyone they deemed capable of doing the job could get the job. At the wrong time and in the wrong city, you would need connections to get in.
Key sources:
Cartwright, Frederick, Disease and History, 1991.
Epstein, S.A, Wage Labor and Guilds in Medieval Europe, 1991.
Thrupp, Sylvia, The Merchant Class of Medieval London 1300-1500, 1989.