There doesn't appear to be any way of checking Straight Talk phone information online in this way.
Unfortunately, it seems to be a service that Straight Talk do not offer, especially not for another person's account.
However, there are some steps you can take to keep an eye on your daughter's phone usage in order to keep her safe...
Is your daughter under 16? If so, ask her to show you her phone and to tell you her password, as a responsible parent you have every right to know what your daughter is doing.
I regularly ask my daughter to show me her phone, I do it when she is least expecting it, and I scan through everything. It's not that I don't trust her; it's more that I love her and don't trust anyone else. I know how bad the world can be, yet she is still naive enough to believe that bad things only happen to other people.
Also, if you are the account holder for your daughter's phone you can request information from the phone company. They will be able to give you a list of numbers called, and texts sent though I doubt they will be able to provide a detailed account of the message content.
If your daughter is over 18 and paying the bill herself, you have no legal right to look at her messages, but maybe you could sit down with her and talk through your concerns.