"My apple is at Tal's."
Translation:התפוח שלי אצל טל!
You can use נמצא here, it's just not necessary. It's a copula used when describing a location. So -
התפוח שלי נמצא אצל טל
is another correct translation.
Yes. But why can’t נמצא (is located at) be used in place of אצל (near, next to)?
Because נמצא doesn't specify where the apple is, just that the following words describe a location. אצל טל means at Tal's, but it could just as well be נמצא על טל (is on Tal), נמצא מתחת לטל (is under Tal), נמצא בטל (is in Tal), etc.