Are you working with any particular certification standard?
May or may not be relevant to you, but I know that BRC doesn't like "production clothing" (i.e. not necessarily food contact) being stored in lockers with personal items, outdoor clothing etc.
You'll know the actual risk best as it's your site, but I'm a little wary as to how well washing will be controlled, and how potential cross-contact with other personal effects will be controlled, if the staff put these gloves in their own lockers. They may not be intended for direct food contact, but would you want them sat in a locker with someone's outdoor shoes that have e.g. mud, dog mess, possibly assorted unknown allergens etc on?
Who does the laundry for the rest of your production clothing at present? In the first instance I'd talk to them about whether you can include the gloves in this. Failing that, washing machines aren't all that costly so personally I'd be inclined to look at buying one for your site.
Nonetheless there are sites where staff are responsible for washing their own protective clothing. Makes me a bit nervous personally, but doesn't mean it can't work
Obviously you'll need to provide a specific work instruction to staff as to how this should be undertaken (e.g. no other cloths in the same cycle, probably the hottest wash the gloves will permit etc), and some means of verifying that it's actually happening through e.g. spot checks. You'll also need to consider what you do in cases where staff may not have laundry facilities of their own.