I would not have a chalazion treated at an urgent care center. I would see an ophthalmologist or ophthalmic plastic surgeon for treatment and/or removal.
This would depend highly on the experience of the physicians. It is very uncommon that a doctor at an acute care facility would have much experience in draining a chalazion. The possible complications that can result if this is done incorrectly are significant. I would recommend seeing an ophthalmologist at the very least. An oculoplastic surgeon would be even better. We see several of these a day. The cost involved would likely be the same since the same procedure codes would be billed, in case that is a consideration.
Acute Care facilities are not the best place to experience ophthalmic care. These locations do not always staff an ophthalmologist. A better option is to call a local ophthalmologist and make an appointment. This is something that may be done in the office setting and some ophthalmologists may be able to provide procedural care on the same day as your visit, saving you time and expense. Best wishes for treatment.
An acute care center will usually refer a chalazion to an ophthalmologist for drainage. Best of luck.