Hi J.,
You didn't mention how long your ex and his gf have been together or if she is there overnight (does she live with him?).
As a divorced mom (who is dating a divorced dad with two girls who I spent time with), I think it is fair to ask your ex to meet his new girlfriend since she is spending so much time with your kids. However, it is not fair to approach it with the mindset that you want to see if she is good enough. Regardless of your history with your ex, this woman does not automatically become the bad guy for being with him or for spending time with your daughters. I would be a bit concerned if your ex hasn't known or been with this woman very long. My boyfriend and I were together for almost a year before we introduced our children.
My boyfriend has two daughters (12 and 14), and they spend at least every other weekend with him. My son and I spend at least one evening with them of each weekend that they are there. My boyfriend also takes my son on early morning fishing excursions (either alone or with his girls). My ex knows about all of this and has no issue - partially because he is less than involved (lives 1000 miles away) but also because he knows I would not have my son around anyone that I did not feel was a positive influence on him. My boyfriend and I do not ever have "sleepovers" when either of us has our kids. I do not live with my boyfriend and I don't do any "mothering" - though I am in the role of an adult friend and they do come to me and talk about things and get advice from me. I see them regularly but I am not their mother and don't try to be - they already have one of those. Their mother and I have not met yet, but I am willing if she ever wants to. She trusts my boyfriend's (her ex) judgment about who he allows around his daughters (he's very protective of them).
It's normal to be concerned about who your children are spending time with - but know that unless she mistreats your children or is a drunk or drug abuser, you have no legal basis for trying to control how much time she spends with them. Ask to meet her - have a frank talk with her about wanting to openly communicate so that way everyone is on the same page when it comes to your girls. Don't treat her like second class because she's dating your ex, and don't be jealous of the time she spends with your kids. Try to see the girlfriend as just another person to love your girls, that's what I did when my ex started dating someone seriously (and we lived in the same state). There are many things in life we can not control - who our exes date (and bring around our kids on his time) is one of them. You are justified in your concern but also need to realize you can't control this situation. She may be a wonderful person who just wants to be a "friend" to your girls (and isn't mothering), but you won't know until you meet her and talk to her.