Whenever you go grocery shopping, just bring it up. Just say "I need tampons. Can we please get some?"
That's what I did. I was nervous to ask but she (my mom) just bought me some and was like "sure, whatever you need". Just remember that your mom is a woman too. She goes through the same thing.
Sydney, there's nothing for you to be afraid of. Your mom has a period too and she has to buy her supplies too. Asking for tampons is not anything bad or wrong. I can't imagine any reason why she would get mad or upset about it. It's not like you're asking her to buy condoms. I'm sure she won't have a problem with it. Tampons and pads are what is used during periods. It's not something to be ashamed of.
I think by talking to her in person is the best way as u build confidence and your mum would appreciate it more if you do. Don't be afraid to do it.
You could either send her w text message or leave her a note so it's less awkward. Just say: hi mum, my period has been really heavy lately, so could you buy me some tampons? :) hope this helped