From what I can see in the Breeder's Manual, you don't have to wait for a specific amount of seniority days.
You can join a team when you get an invitation from one. Most of the time, you'll have to reach out to teams and "apply" yourself for that to happen. Since teams aren't visible on the app, use the Classic version in your browser, hover over Achievements on the far right and click on "Best teams". That will show you all the teams ranked with the best on top. You can also select horse teams or unicorn teams. Once you've chosen that, you can click on a breed you're interested in in the list on the left, which then displays all the teams for that breed ranked with the best on top. From here, you have to click on each team and look at their presentation page to see if they're recruiting and if so, what the requirements are.
Once you've found a team you find suitable, you reach out to the person they tell you to reach out to on their presentation page.