If breasts are removed due to breast cancer, can a woman get new breasts via an implant?

Yes: Yes. Typically a bag will be put in under the muscle at the time of the breast removal (mastectomy) and then salt water put into the breast every week until you get to the size you want. Then the bag (expander) is taken out and a saline or silicone implant is put in.

Implant - yes: Implant reconstruction is commonly done for breast cancer. It is sometimes done at the time of mastectomy. Not sure what you mean by new breasts. Once removed, you will not have breast function (lactation, nipple changes...) even with the best reconstruction.

Yes: Yes they can. There are also other techniques where tissue from different parts of the body are used to reconstruct the breasts.