I have used this method for wood plank flooring.

Have someone stand on the gap of the two boards (they should be ensuring the two boards are level with each other.)

Take a pry-bar (crowbar) and push the flooring from the edge (may have to remove quarter-round that is covering it).

Basically you are pushing the entire floor to get the boards to snap together. The person standing on the gap helps to ensure the boards are level with each other and prevents the other side of the flooring from moving. It can take some trial and error to figure out. You want enough weight on the opposite side so that the planks will snap together.

I haven’t worked with vinyl planks so not too familiar with them. With the wood ones you had to be careful not to break the tongue sections.