This is a difficult question to answer.
Mechanically, something like the Imprisonment spell referenced in Is there any way to trap someone's soul? would work and other ways to trap souls are also talked about.
It shrinks the target to about one inch in height and it describes them as being stuck inside of a gem.
As to "not being able to escape" that will always depend on if this entity / person / demon has any friends who could free said entity. If you keep the gem / prison on your person, then you'd have to be overcome before this entity
could be freed.
You'd want the same hope of some chance of being freed if it were cast upon you, I suspect.
I suspect, without further details, that you already have a way to collect and deliver souls though since that is what the Raven Queen is asking for.
Beyond that, the short answer is: No, you can't deliver their soul without killing them.
However, it is really up to your DM and his interpretation of what would please the Raven Queen. It could be that something like having them sign a contract that on their death their soul is hers would be enough.
I seriously doubt trying to trick a goddess with a copy of a soul would work very well. You could possibly negotiate with her with other souls? Think of something like what Jack Sparrow does in the Pirates movies when he barters for his soul with Davey Jones.
However, if he expects you to literally reap the soul of a fellow party member, I suspect that he has some different and interesting plans. It sounds like your DM put you in this situation to cause some conflict in the party on purpose and is waiting to see how your character would react and what they would do when given this kind of choice.
You may want to ask your DM what his intentions are. If he expects this to be an interesting avenue to roleplaying opportunities and seeing how far your character will go to serve the Raven Queen, or possibly if he has spoken with the player of your target and intends you to go through with it in some fashion because the other player is ready to start a new character at the table.