Can a US citizen or anyone get VAT reimbursements when living in Thailand with a retirement extension?

Perhaps an accountant or someone who tried can answer this question.

The apartment charges VAT and it adds up over time.

No you cannot.

A Thai company that is registered for VAT may offset the VAT from purchases against the VAT from sales and reclaim any difference. This is an advantage to manufacturers and exporters in particular, as VAT is not charged on sale of goods to customers who are outside Thailand (goods that have proof of shipping), but the company probably paid VAT on raw materials.

As an individual living in Thailand you have to pay VAT like everyone else, regardless of nationality.