What are my (unreserved) accommodation options when doing a road trip in the US with (small) children?

Hotels along the major highways and interstates are plentiful. My personal recommendation, if you'll be traveling with a smartphone of some sort, is to pull over at a rest stop an hour or two before you intend to finish driving and check out what your options are within 50-100 miles. Most of the major chains have good mobile sites/apps, and free wifi is relatively common at many stops these days as well.

I'm from England and work for an airline (i.e. air travel is cheap and easy for me). We've done fly-drive holidays in the US and Canada at least every other year since the kids were born. Pretty much every town and city in the US has motels and hotels on their edges. They're mostly OK. Some are excellent, some not so good. For a family with small kids a room with twin beds is probably the cheapest option. In my experience they're always double beds but check and see the room before accepting it. They never have a problem with you doing that.

Walk-in prices are the most expensive. If there's a phone number displayed, phone them for a price even if you're parked nearby. If you do just walk in, don't accept the first price offered. Say you'll have to check with your partner in the car and they'll often throw in a discount to stop you walking out.

Check what breakfast, if any, comes in the price. Often just coffee, perhaps a small sponge cake or similar. Occasionally a buffet.

As many people have said, cheap family-friendly lodgings are plentiful near interstate highway exits and other major roads. (In the US B&Bs tend to be pricier luxury accommodations; look for motels. Off-brand motels can be even cheaper but service levels can be spotty. You are unlikely to check into a chain motel in January and find that the heat doesn't work, for example.)

However, it is well worth getting reservations in advance -- even a few hours in advance -- as you might be surprised to learn what areas tend to sell out their lodgings either regularly or for some obscure local event. You will also usually get better prices booking online than by walking in the door.

If you have tents with you you can also consider some of the chain campgrounds (or other local campgrounds, but they are going to be harder to find online), but beware, in the US a "campground" is often primarily a place to park your RV for the night and may not be a friendly place to pitch a tent.