Synonym for "on-the-job" as in this example

Depends a bit on the context and your intent. There are two common ones in professional contexts.

If this fragment is part of a description of the necessary qualifications for a job, and you want to tell applicants that they must have worked a similar job in the past, then the usual format would be something like "To be a lab tech, prior professional experience is required." This specifically requires experience in a formal position - usually paid - working with the requested skillset.

If you only require the applicant to have worked in some kind of live environment (as opposed to only receiving training) then it would typically be "prior hands-on experience". In this case, informal jobs or practical experience during education is considered sufficient.

I think "To work as a lab tech, prior work experience is required" would be a bit strange, but I do see "Prior work experience" listed as a bullet point in lists.

Generally, "on-the-job experience" isn't used to describe a required qualification, but rather something one might gain - e.g. "Our co-operative education program offers college students a chance to gain on-the-job experience while still in school."

"First-hand experience" generally refers to having experienced some event in person, rather than job experience - e.g. "We are looking for contributions from readers with first-hand experience of a bear encounter." It would probably be strange in this context.

"On-site experience" generally refers to experience working in a locale where you aren't necessarily required to be (usually closer to the customer or otherwise beneficial for your development) during a job. For example, "Sally gained a lot of valuable on-site experience when she went overseas to work with the client at their office in Germany, rather than staying at the home office." It wouldn't be necessarily incorrect here, but the meaning would differ.

"On-the-job" is an idiom mostly associated with training, not experience. OJT is usually given to those with little experience in entry level positions. Positions requiring experience typically use wording that uses the word experience unmodified.

To be a lab tech, experience is required.

To be a lab tech, prior experience is required.

To be a lab tech, previous experience is required.

To be a lab tech, prior work experience is required.

To be a lab tech, previous work experience is required.

To be a lab tech, related experience is required.

Job announcements which include this as a requirement usually quantify with a number of years required. Primarily, they want someone who has done the job before and only needs to learn local policies and procedures, not the whole job. These jobs usually pay more than entry level jobs with the same description.

Plus all the ones Backgammon listed.