Do dental hygienists order things like toothpaste and other equipment?

For their job like in the office (:

Yes, the hygienist order what they need for their prophy patients, the office orders what they need and the assistants usually order their dental materials.

In most offices it is the way I've described, although every office is different, I know of several others that have one hygienist that does all of the ordering for the hygienist and the assistants. This is done by asking the assistants what they are out of or what do they need. In all of the offices I know of, the front office always takes care of their own since their supplies are on a totally different spectrum from what the hygienist and assistants needs are.

As for the responder who is a certified assistant with x ray experience/training that could save her doctor a lot. It doesn't matter what training or how much experience you may have to be given the authority to do the ordering. Your doctor is going to give that authority to the person that he trust, who is familiar with the materials he likes to use and who knows what it takes to keep his office running smoothly. Your doctor may not be so concerned with saving money when it comes to taking care of his patients, no matter what you think each of his patients hour is worth.

do you mean for their office.. or personally?

In some offices they do.
From what I've found, office's that have office trained demtal assistant's make the hygienist do most of the work.
I am an assistant and I am used to doing the ordering for offices, but I have recently transfered to a new job. This dentist is used to training his own assistants and I am certified w/ and x-ray license. He doesn't allow me to do the ordering (which ticks me off). The hygienist does all of our ordering and her own.
Personally, I would ask for more money if you are going to do the ordering and cleaning teeth. The dentist is already making alot of money off of you.
Us assistant's aren't paid as well as we should be (considering every pt. is worth atleast $200/hr.)
If you are not a hygienist and just pondering, most of the time, the dental assistant does the ordering for the office.
It really depends on the office and the situation.
In my case, I am a well knowledged assistant and know alot about products and prices of products. If the dentist I worked for allowed me to order, I could save him a bundle! But, he opts for his hygienist to do so. Sucks for him.... :P