You have some say in it, but they tend to like doing PICCs until it's not really an option, because a port is more permanent. I got mine at 18, after having had only 3 rounds of IVs. But this was because PICCs only gave me trouble. Every line I had was pulled before the meds were done because I got phlebitis (irritation of the vein, causing the line to be pretty useless).
If you think it's about time and you'd really rather look at a port, ask people. If they don't know what you're talking about, run the hell away. Any nurse who doesn't know what a port is frightens me. That's like asking a nurse about an IV and having them go "Huh? What's that?" Frickin ridiculous. In short, if you ask about it, and they don't know, go to someone else. If the new one doesn't know, go to someone else. Just keep moving on until you find someone who knows. It's a common enough medical device, you WILL find someone who knows what it is.
The one thing to keep in mind with a port is that it can get in the way (sometimes -- contact sports is one of the ones it lists, I believe), and it requires accessing once a month, whether you're using it or not. It has to be flushed with Heparin, to make sure the line doesn't clot. Once a month, every month.
I just wanted to point those out. Having said that, I would recommend a port without a second thought. I LOVE mine, and it's totally worth the little bit of permanence and maintenance it requires.