How do I ask my dad to buy me tampons/pads at the store without making it suddenly awkward?
The first couple of times, it’s going to be awkward no matter what. Fortunately for you though, asking will get easier over time. Buying tampons/pads at the store with a male cashier is embarrassing too; yet, everyone who has to buy these things just has to get used to it, and with time it becomes more comfortable. Remember that there is nothing about which to be ashamed. Realize that many girls and women have to buy these products every month just like you.
Since your dad was married to your mom, he probably picked them up for her at the store for her from time to time; it is possible that he has bought these things before! Even your dad may not feel as uncomfortable about it as you may think he would, so the best thing is just to ask, to start. Break the ice! Once you ask the first time, and then the second, then third, and then the fourth, such requests will start to feel more normal. Maybe you guys will work out a system where he gives you money for them and you pick them up, or he just buys them and stocks the closet. Who knows, but you have to start somewhere!