A family member is beginning to plan her bridal shower, and would like to invite approx 50-75 people. Based on that, it would need to be at a restaurant/hall.
Based on this statement, that family member (it sounds like this is the bride, based on 'her' bridal shower) should be paying. Whomever plans: decides on the location, decides on the number of invites is the one who pays.
In my social circle, it would be incredibly rude for a bride to plan her own shower (generally, the couple, or the bride's parents throws an engagement party- you can throw your own engagement party, for some reason it isn't considered the same thing).
If it is expected that the bridal party foot the cost for the shower (and that is very common), then the bridal party would decide where to have it and who to invite. I've been in weddings where the ONLY people at a shower were members of the bridal party, because that was what the maid of honor could afford to host.
It isn't uncommon for mothers, aunts, grandmothers, church groups, work friends, etc to host bridal showers either, or for there to be multiple showers. Multiple showers is often so that they can afford to be hosted- each shower has 20 or so people at it, not 75!
If I were a member of a bridal party, I would NOT pay for a shower that was planned (and extravagantly planned!) by someone else. Most of the shower's I've seen have been 15-30 people in a home. Not 75 at a restaurant. (Which is part of the reason for multiple showers. Having 75 people for a meal is basically a wedding without women bringing their spouses or children.)