In August/September you are can order an annual replacement white coat at no cost to you (though you might have to pay for the embroidery). If they are asking you to pay for embroidery, I think it is around $12.
Please submit the Lab Coat Order Form to email@example.com and firstname.lastname@example.org. Their phone number is x33985. When you submit your order form, I would speak with the linens staff to confirm that they will not be charging you for the coat.