Can I just send my customers an SMS?

Not just like that. For commercial messages the rule in Belgium is that you need consent, with one exception.

The exception for existing customers

To anyone who already bought from you, you may send messages about your own, similar products — provided you obtained the number during that sale and every message contains a way to opt out.

For everybody else

Prior consent. Ask for it explicitly and record when and for what it was given: in the event of a complaint you must be able to demonstrate it.

What is always required

Your identity must be clear — a message without a sender is a problem. And opting out must be possible, in practice with STOP. That request must be processed immediately.

What falls outside the rules

Transactional messages are not advertising: an appointment reminder, a shipping notice, a verification code. You may send those without marketing consent, as long as you sell nothing in them. Add an offer and it does become advertising.

Buying numbers is not allowed, and does not work either — such lists mainly produce complaints.