When we find shoes we like on a big discount, is it a good idea to buy a few sizes up at a time (like sizes 7, 8, and 9 all at once)? Or is there enough overlap in sizing that our toddler might move directly from size 7 to size 9, skipping over size 8 as her feet grow?
I would only purchase what is needed now, for a number of reasons.
The rate of a child's growth is not linear. The child might take stay at size 5 for 8 months then be at size 8 2 months later. Or the feet might not grow but the child gets taller.
What you like now you (or the child) might not like in 6 months.
My home had far too much kid stuff already, without stocking up on this kind of stuff.
If it is on sale today, it will be on sale again later.
I found that my kids were not great at telling me when shoes were to small and I did find that they made big jumps, so I wouldn't buy all the sizes together. I also found that Target and Costco often have sales on shoes for young ones, so I wouldn't worry about not being able to find shoes when needed.
I also pass down shoes through children. As they get bigger they get worn out and can't be passed down, but at that age save them for the next one!!!
Realistically, toddlers will not need every size of shoe. The difference between whole numbered sizes is only 1/3 inch, and half-sizes are only 1/6 of an inch different, so little you might not even notice the difference, especially depending on the type and fit of the shoe, one of the reasons some shoes (eg: flip flops, crocs, rain boots, etc) are often manufactured at size intervals greater than a whole size (eg: sizes available are 5/6, 7/8, 9/10).
With the space you are supposed to leave in the toe of the shoe for the foot to move, by the time you realize the shoe is too small and hard to get on, it is likely the jump to the proper size will be at least a full size, if not more, especially if you try to get the largest shoe your child can walk comfortably in without tripping or stumbling to maximize the shoe's longevity when you get new shoes. My son is almost 2, and so far on average he has gone up about 1 1/2 sizes each time we need new shoes for him.
You could potentially use shoes of each size, but overall you would end up with more shoes than necessary that way, so whether that would be worthwhile would depend on how good of a deal the shoes are and whether the quality of the shoes is high or not.
It also does not seem like the feet grow linearly over time, but much like height, they do most of their growing in spurts where they grow slowly for some time and then grow quickly for a shorter period of time and so on...so you may find the amount of wear you'd get out of every size would be uneven.
Also, feet are not always the same size, often one foot will grow slightly faster than the other, and often the feet will be half a size different (which is why you measure both feet!), so there may be some slop room as to which shoe size fits best, and there may be a small range of "those ones fit ok but would get outgrown fast, those fit ideally right now, and those seem slightly loose but will last longer" leading to choice in the ideal next size.
Having shoes on hand at various size intervals does have some advantages though. For example, if you child suddenly has a growth spurt, you would not be caught off guard one day by finding the shoes are just so tight they are painful to put on and needs new shoes immediately. You could, instead, pull the bigger shoes out of the closet. However, this also has it's drawbacks that a particular shoe might not be suitable to the season they fit properly. In the summer you might want a sneaker with air-vents on the side to keep the foot cool, or sandals, but in the winter you might want rain-boots, or warmer sneakers with thicker insulation. So if the deal is really unbeatable (eg: size or two up shoes at a consignment sale perhaps), it doesn't hurt to get larger shoes, but if you would feel cheated if you should overlook the window of time those shoes fit, or they fit in the wrong season and you don't get much use out of them, you might wish to pass and just get what fits now.
I have found excellent deals on toddler shoes at consignment stores. I just got 2 pairs fo Stride Rite shoes (one looked new, the other slightly used) for $6.50 total. There are great deals to be found all the time.
If you find a REALLY great deal on some really basic shoes that can be worn year round, I'd buy several sizes up. I'd only recommend this if you know it's a good shoe (maybe a brand you've bought before), and it's a good enough deal that you'll still feel like it's a good deal if your child never wears one of the sizes. Although, I think having the shoes sitting in the drawer might make you move up sizes more readily than if that involved going shopping for new shoes, so this might help you avoid skipping sizes.