Be sure to make an alternate dummy email account when applying for jobs so all the advertising agencies the company sells your contact information to for cash will send their spam towards an account you rarely access.
Employers are incredibly protective of their candidate databases; you will not have this issue when applying directly to a company. They're not out there trying to turn their recruiting department into a profit center by selling email lists. If you've had this issue, it'll have been from job search sites like Monster that are trying to monetize their business as much as possible.
OP, SSN is typically used to do background checks and, depending on the job, credit checks. It's not usually collected up front during the initial application but the retail industry is an exception to this. Because of the volume of candidates they handle and the drop-off rate that they get if they come back and ask for more info later, they'll usually try upfront to get everything they need to process you all the way through to hire.
Is there a risk? Eh, no more than average. If it's a paper application you're depending on the people handling it to treat it securely, and that's probably hit or miss. If it's an online application you're a little better off but you're still expecting the software provider and configuration choices of the employer to treat it securely, and honestly that can be hit or miss too. You're unlikely to know either way though, and it's an unavoidable piece of life and work at the present time.
<------works in recruiting software.
FWIW, that Walgreens job site looks like a homegrown applicant tracking system.