Some MSPs will let you defer providing your SSN. After completing the sign-up, you can check their job board for a few months to see if they're a MSP you'd like to work with, then provide it, although I think the IRS is not allowing that anymore. For those MSPs that have a secure website, I provide it.
A few times, I'm leery. I've entered numbers, but not mine and not a valid SSN - like 999-99-9999 or 123-45-6789. It may work, or what's funny is when a prompt says the number has already been used. There have also been MSPs without a secured site. Two that I can recall have emailed and/or telephoned me and we've worked it out satisfactorily.