Typical rule of thumb: the smaller the container, the fewer flowers needed to fill it. But be careful - a lot of people look at a vase that they think looks "small" but it is actually just short. The size of the opening is what you need to look at.
The vase here is short, but it has a wide opening, and therefore needs a lot of flowers to fill it (of course, depending on the design you go with, this can be adjusted... but in general)
Here are a few ideas for vases with smaller necks:
Eiffel Tower/Maestro vases
These vases are tall and very narrow, requiring only a few stems to fill them and create a simple, elegant look, and provide some nice height.
These little vases only need about 1-3 flowers to fill them, and look great clustered together on a circular table or in a row on a long table. You can find a wide variety at thrift stores, craft stores, etc. and mix and match the styles for a unique look.
There is such a huge variety of options when it comes to vases/containers for centerpieces, but if you're looking to cut back on costs without cutting back on the number you need, look for some containers that won't need as many flowers to create the look you want.