write a little list and give it to your dad saying, "dad, i've run out of a few personal supplies, mum used to get them for me but could you get them for me next time you go shopping?"
then give him a list something like this -
deoderant,
spot cream
tampons
cotton wool
etc
i guarantee he won't be embarrassed and i guarantee that he already knows you have periods and use tampons (he has looked in the box, wouldn't you have?), he is just not thinking about it because of losing your mum - he has a lot on his mind. it is perfectly natural that this wouldn't have crossed his mind.
he would actually probably be a little upset if he knew how you were agonising and suffereing over this because you were too scared to tell him
or ask an auntie or someone to tell him