Technically incorrect. Bluetooth does not have MAC address. It have addresses named BD_ADDR, not MAC. It is also a 48-bit value, like Ethernet MAC (Media Access Control) address, but it is different thing and have different structure. MAC address contains Unicast/Multicast, Global/Local bits in MSB, vendor ID and adapter ID. BD_ADDR consists of NAP (Non-significant Address Part, used for frequency sync, it could change) , UAP (Upper Address Part, seed for encryption) and LAP (Lower Address Part, device address itself). Google it if want to know deeper details.

When you scan for Bluetooth devices you see Bluetooth addresses, not MACs.

To see MAC addresses you have to scan for WiFi packets device exchange with local WiFi Access Point, f.e.

Also, as some *winner mentioned, it is barely possible to unnoticeably implant BT device into human body due to size and power demands.

However, IIRC there was some efforts to push mobile apps that show proximity to vaxxed/unvaxxed persons mobiles, and that really could be used to check if there are any vaxxed around to some extent. You need to reverse engineer that app to find a way it obtain vaxx status information for Bluetooth addresses it find around. I think it is some requests to an internet server with a database.