I've run SQF programs where we didn't require employees to wear anything over their street clothes in warehouse/storage areas. Defense was no product is allowed to be opened or sampled in warehousing, so all food is kept in sealed poly bags in boxes and is safe from contamination risks, plus our GMP program stated employees must wear clean clothing from home to work. Since you mention it is a distribution area, I assume the same logic stands.
Since you're providing the clothing to the employees, it stands to reason you'll need to develop a regular laundry schedule for it. Any time an auditor can spot the clothing is dirty from day to day use, they'll have an excuse to make a finding. I'd start by getting the coats laundered, then observe how quickly they become visibly dirty through normal employee use. You might find a weekly or bi-weekly laundering is effective that keeping them clean enough for this low risk area.