PhD programs can be highly individualized, and you can theoretically pursue this from any program with the right advisors and resources in place. That said, what you want sounds more like a degree in computer science with a focus on educational technology.
But let’s set all that aside, because you absolutely do not need a PhD in computer science to develop educational technology. What you need is a computer science degree and some internships, hopefully at edtech companies.
Having a research internship the summer after your freshman year isn’t super common; freshman usually haven’t completed enough classes to be helpful in the lab. That doesn’t mean you shouldn’t try, though.