there are many on the ground in Thailand and as suggested check out the Facebook site as suggested and Thai PBS which is an English news service. Am sure your family would know what is going on so long as they have access to normal media
Look at Facebook pr Thailand Government there are videos and posts about vaccine rollout .For the majority of the population both foreign and local the roll out begins June 1 .At the moment priority is for vulnerable and front line .Perhaps your family members would have been seeing information on their news bulletins as the majority of the population do .
At the moment only front line and vulnerable Thai people are being vaccinated in Thailand. There is either a shortage or no supply of certain vaccines.Suggest you make further enquiries through Thai Government sites