It depends on the union, the industry, etc.
In a lot of cases, a few disgruntled workers start talking over lunch or something and decide that they’d have more leverage with their employer if they organized. They then go out and seek out a union representative to talk to (usually, they either contact the most logical fitting union, or else someone has a friend in a union shop elsewhere and contacts their union. That’s how you sometimes get wacky combos like the Communications Workers of America representing nurses or Teamsters representing teachers). A union representative will set up a series of meetings at a nearby restaurant or bar or at one of the disgruntled employees’ homes, and encourage the disgruntled employees to talk it up and get their fellow workers to come listen to the union’s sales pitch and ask questions. Depening on how flammable the situation is, the union might also start leafletting employees conspicuously or otherwise making its intentions known. The goal of the union is usually to get employees to sign authorization cards, indicating that the employees want the union to represent them. If the union can get a majority of these cards signed, it can either present them to the employer and demand that the employer voluntarily recognize the union, or else it can present the cards to the National Labor Relations Board and demand that a representation election be held.
That’s the tpyical model-- the kind of thing you see in movies like “Norma Rae.” Some unions will proactively focus on a particular employer, and seek out disgruntled employees there, rather than waiting to be contacted. (Often times, both methods are at work at once.) The construction industry, given the short tenure of most construction projects, operates a little differently. There, the construction unions typically train their members in particular trades like carpentry and plumbing. Since the union controls the supply of skilled tradesmen, construction contractors usually form standing affiliations with union locals. When the contractor is awarded a contract, it enters into an agreement with the union to use only that union’s members for the project, and the union send the contractor the requested number of tradesmen-- there’s never any union election.
Looking more closely at the question, though, it appears that you may be talking about a situation where the union is already in place. In that circumstance, the union collects gripes from its workers about unfair treatment on an individual or collective level, and takes action as it deems appropriate. If enough workers complain to their union steward (usually a fellow employee who assisted the union in its organizational effort) about a general policy, the union will negotiate with the employer about changing the policy, either during the next round of contract negotiations or immediately if the policy is amenable to being changed under the existing contract. The union will also take personal grievances to the employer for adjustment-- typically, almost every discharge of a union member will result in a grievance being filed by the union with the employer. It’s unusual that the union requires anyone to write a letter initially, since the union stewards usually work alongside the other members, and is usually aware of what is going on in the workplace anyway.
The unions also take it upon themselves to initiate proposals despite not receiving any requests from the employees about them, but these proposals are often ones to preserve the union’s own position, such as propsals requiring all employees to join the union within 30 days of hire or a provision allowing union dues to be deducted from an employee’s paycheck (instead of the union having to shake everyone down for their miserable $30 a month). And, of course, the union generally does its own research about wage rates in comparable industries to develop its wage proposals, without necessarily having to survey the employees to ask how much of a raise they think they should get.
Ummm… how’s that?