Building a form does not send it anywhere. Until you assign it, it exists and nobody has it.
Assigning
Open the form's assignments and you get your client list with a switch each. Turn on everyone who should receive it, then Apply. Basecamp's own instruction for the other direction is equally plain: switch off anyone who should no longer receive this form, then choose Apply.
Nothing changes until you apply, so you can work through a long list without half-applying it.
With no clients yet you are told to invite some first. You cannot assign a form to somebody who has not been through onboarding. That is what public forms are for.
The first one arrives straight away
Whatever frequency you set, the form appears on a newly assigned client's dashboard immediately. The recurring timer only starts after their first submission.
So assigning a quarterly form does not mean waiting three months to see anything. Assigning several forms at once puts several on a client's dashboard on the same day, which is worth staggering.
What the client sees
The form appears on their dashboard and stays there until they complete it. Nothing stacks up. Miss one and it waits, and the schedule restarts from the date they submit rather than from when it was due.
Everyone runs on their own clock
Scheduling is per client, timed from when each person last submitted rather than from a shared calendar. Assign the same weekly form to ten clients and you will get responses spread across the week rather than all on Monday.
That is deliberate. It does mean that "has everyone checked in this week?" is a question you answer by looking, not by assuming.
Unassigning
Switching a client off stops them receiving it again. Responses they already submitted stay. Unassigning is not deleting, and their history is intact.
This is the right way to retire a form: unassign everybody and leave it alone. Deleting the form destroys every response ever given to it.
Assigning from the client instead
You can also work the other way round: open a client and manage which forms they have. Same result, and the better route when you are setting somebody up rather than rolling a form out across the roster.
Building a check-in form covers the other half of the job, including the schedule and the unique name you cannot change afterwards. Reading form responses covers what comes back, and what an empty list means: a form nobody was assigned to collects nothing at all.