These two sentences are very similar, and a lot of times, they can be used interchangeably. However, there is a slight difference. When you say "When will I meet her?", you are just asking for a time in the future when you will meet this girl. When you say "When CAN I meet her?", you are asking "At what time will I be ALLOWED or PERMITTED to meet her?" It's also another way to say "At what time will I BE ABLE to meet her?" So there is a difference between the two, but that difference is very slight and doesn't really affect usage at all. In most situations, you can use either of these two sentences, and the person you're talking to will understand you perfectly. One or the other, it doesn't really matter. But maybe someone else will explain it better than me.

The construction 'get to do something' means 'have the opportunity to do something'. In this situation, the speaker wants to meet the girl but has not had the chance to do so yet. A common use of 'When do I get to meet her?' is when you know that a close friend or family has a new girlfriend or even fiancee whom you are anxious to meet. If your mother, for example, asks you about your girlfriend and says 'When do I get to meet her?', she means 'I want to meet your girlfriend. Why haven't I met your girlfriend yet?' and 'Why haven't you brought her home to meet me?' The implication is that you are perhaps preventing your mother from meeting your girlfriend. This is a very common construction. Another typical use is this: 'I went on a business trip to Paris, but I didn't get to do any sightseeing'. This means that I wanted to visit the sights of Paris, but I unfortunately didn't get the chance because I was too busy : my work prevented me from going sightseeing.

The 'can' part will emphasise more on your interest to meet him/her.And 'will' part shows that you are waiting for someone to meet and asking someone about his/her availability.