I have had italki messages from Chinese people who have used HAHA to indicate laughter, personally I use LOL (laugh out loud) in my texts.

As you are from Japan, the culture may be different. In other places, ladies do use haha.

I think it’s fine.

It's not weird at all for females to use it. "haha" is a gender neutral expression you can use to express laughter or even sometimes mockery. My female friends and I use it all the time when texting.