Why dont you make T-shirts part of the uniform and get the employees t-shirts with the company logo on them?
With bringing clothes from home, you have to make sure their clothes are clean and also if they get soiled at work, you need to be able to have them change into something clean so they arent walking around with soiled clothes. Also, if you have a 3rd party currently doing laundry, they wont be cleaning the street clothes and that could be a problem.
So far, the company I work for has danced around the issue of having our laundry done by a 3rd party, we issue out company t-shirts and pants to our employees and they are responsible for keeping them clean. They can get a replacement if they show their old shirt is worn or messed up by bleach etc. We have a washer/dryer available for employees to use, but no one uses it. They all take their shirts home and wash them.
We have been told by SQF auditors that if we are not having our laundry done by a 3rd party, we need to have a laundry program. What the laundry program would entail, I am not 100% sure but I would imagine it would have to allow cleaning on site (washer/dryer available to employees) and also show that the cleaning method (the type of laundry soap you buy vs the typical dirtiness of the clothes when washing) is effective.