Let's first deconstruct your sentence into two independent clauses and worry later about combining them by subordination.
The Member State is responsible for processing the asylum claim.
The asylum seeker enters the European Union ________ a Member State.
The preposition from doesn't work because, while one might enter the EU from a non-member state like Turkey or Morocco, once an asylum seeker has arrived in a member state, they are already in the EU. One cannot enter a place where one already is.
Instead, one enters the EU through a member state, say Italy or Poland, even if the ultimate goal is France or Germany. This yields:
The asylum seeker enters the European Union through a Member State.
Subordinating this sentence to the first as a relative clause modifying the noun phrase the Member State yields:
The Member State through which an asylum seeker enters the European Union is responsible for processing the asylum claim.
There is no need for stranding the preposition in the construction which the asylum seeker enters the EU through. This is fine for informal conversation or writing with different vocabulary and a lower register but not for formal writing.