I have seen people make videos and pictures where their dog looks a lot wider and bigger in the video/pic than in person. What kind of camera lens are they using?
It's a matter of perspective. You just shoot the dog at very close range, while making sure there is some background visible (this can require a wide-angle lens, but can be done with a smartphone).
The hard part is keeping the dog reasonably still while you shoot, and avoiding tongue marks on the lens.
For exaggerated proportions, you need a wide angle lens (or setting). Avoid a fisheye or vignetting since it makes obvious what happens. The relations a wide angle lens exaggerates are the size increase as things get closer.
A semi-frontal perspective is dangerous since it increases the head size compared to the body, leading to "baby" proportions. You'd either go full frontal or take a side view. That gives basically a "flat" perspective where the visible parts of the dog are all more or less the same distance from the lens. It is not untypical for zoom lenses to have a bit of barrel distortion at the wide setting: that also helps in getting a larger impression.
What breed is the dog ? Different lenses will have different effects depending on the breed and whether the dog in question is good or not.
The Hundplotzer Canigon design is known to work well on generally agreeable Terriers but it is said that Max Berek himself spent the last decade of his life in frustration trying to design a lens that could deal with belligerent Dachshunds.
And of course some breeds like the Labrador are inherently unphotographable without special filters because of a mutation that prevents them from reflecting light above the near infra-red - this often makes their owners resort to alternative methods such as 3d-rendering their dog or hiring sheared Golden Retrievers as stand-ins.