Would families like the Starks or Tully’s normally have a decent number of their bannermens children as wards, and if not, how do they determine who ends up fosters normally?

Depends on the perceived need for strengthening ties between the houses, but it seems fairly common…if we skip a generation either way. 

While not precisely wards, the fact that Brandon had Ethan Glover, Kyle Royce, Elbert Arryn, and Jeffory Mallister as his companions makes sense: it shows favor to the Glovers although that’s more a personal connection, but it’s also about tying the next generation of the Vale and the Riverlands and the North together. 

Likewise, while Robb only had Theon growing up, once he became King in the North, he’s got a good selection of bannermen: Umbers, Manderlys, Flints, Mormonts, Norreys, Karstarks, Hornwoods. Plus when he heads into the Riverlands he picks up Mallisters and Freys to add representation for that kingdom.