Will you be starting the job before you get a clearance (i.e some of the work doesn't require a clearance)? If so, they may be waiting to have you fill out the paperwork until you start working. But, if you need the clearance to start working then they should have at least given you the forms to start filling out. I'v had several jobs that required a clearance (higher than public trust though, and I'm not sure if that makes a difference), and they all sent me the paperwork or a link to the paperwork within a couple of days of signing the offer letter.
If it were me, I would call the HR person who you've been in contact with them and ask them about the expected timeline.