The short answer: Both LVP and carpet are very low-emission, low-VOC products. Carpet probably has a little bit more of an odor than LVP, but not much, and it goes away relatively quickly.
New carpet
There's a tiny bit of an odor with new carpet that goes away relatively quickly. Years ago, they used formaldehyde in the adhesives that glue the primary and secondary backings together, and it had a strong odor. A lot of people have a memory of that strong, strong new carpet smell. That was the formaldehyde. They long ago removed it, and it's now gone from the adhesives.
From the Carpet and Rug Institute: "Formaldehyde is not used in the carpet manufacturing process. It is not emitted from new carpet." They also say "most VOCs dissipate within 24 hours—even faster with good ventilation." (source)
New LVP
I've never had anybody tell me there was a real odor to LVP. (The IXPE foam backing on most of our LVP is low-odor and low-VOC. See What to Look for in LVP.)
Questions about a specific product? Come see us in Provo or Lehi.