Call stored procedure via trigger or SQL Server Agent
To answer your question directly, the trigger will not finish until the SP finishes, meaning your DML statements that cause the trigger to fire will also not finish (until the fired trigger finishes). This may cause long duration of open transactions with locks everywhere.

So including SP inside trigger is a big NO. This can cause huge performance issue when your SP slows down for whatever reason. It has direct negative impact to your workload throughput.

Not sure about your sql server version, if your version is SQL Server 2008+, you may consider Change Data Capture as your solution.