Theres no way to know you need the "ami" from the english. This is a bad translation.
"which is under the big table" would make much more sense for English.
Could the English translation here be "that which is under the big table"?
can we take out van?
No, we cannot. "A nagy asztal alatt van" is adverb-like, & it acts like an adverb, so you need van here.
"Az a nagy asztal alatt van." should also be correct, if the sentence would start with a capital letter, where "az" would mean "that particular thing".