@ Pleasance:
One of the few times I got transfused, the nurses had a very difficult time starting the IV for the transfusion because, apparently, my veins kept wanting to "collapse". The initial poke went okay, and then when they were going to flush and/or draw some blood before the transfusion, the vein collapsed around the needle and they couldn't proceed any further.
I was told at the time that it wasn't a serious problem - more of an inconvenience than anything else as this meant I would have to get poked more than once - and that having low blood pressure and being dehydrated can make the veins more prone to "collapse".
The reason for the "collapse" which results in either blood being unable to be drawn or the flow in an IV being unable to flow is usually because of the vein's response to pressure against the wall of the vein and for more sensitive/delicate veins, the suctioning pressure of a needle can prompt the vein to "collapse" around the needle.
I have low blood pressure and am easily dehydrated, so when I heard that my vein collapsed on me multiple times during that time I wasn't very surprised. Furthermore, I am considered a "difficult draw" because it has never been easy to draw blood from me or start IVs even without problems related to my veins "collapsing" simply because I have small veins.
A larger vein is more promising when it comes to both venipunctures (blood draws) and IVs (including for transfusions) because in general, they are less sensitive to pressure/changes in pressure when they are being punctured.
Perhaps your daughter (I believe you are talking about your daughter? If not please correct me!) is experiencing similar issues that I did?
If so, and this is just my gut feeling, I don't think you need to worry so much about the problem.
Drink lots of fluids before the transfusion and/or before blood draws and be patient as the nurses/doctors work with you and your daughter to find the best site for the transfusion.
For myself, the inside of my wrist (closest to my thumb) has been the best place for an IV if the inside of my forearm (bend of elbow) will not cooperate. The top of my hand also works well, usually, but the inside of my wrist closest to my thumbs seems to have worked the best.
Also, I have been told by a blood draw specialist that the veins in the dominant arm (the arm that is used for writing - right handed, left handed, etc) are usually "stronger"/less prone to collapse compared to the less dominant arm because the muscles of the veins have had more exercise.
Perhaps this is something to be taken in consideration as well?
All the best!