As the previous answer said, you should think about the performance issue. And responding directly to your question about "Or is this a job for a background service?", I would say it would be advisable to do so.
Think about the problems that could be caused if you entered 10 records, and suddenly 3 of these email sending processes hang. Or even think about a scenario, if you were using a gmail SMTP for example, and this request took about 15 seconds to be resolved (as has happened to me in some cases). This would most certainly affect the performance of your database.
I would think about another question: it is not my database responsibility to send an email. So, I would do this either through a command line or, as already suggested, using an application's backend to do this.
As I suggested in my comment, I might consider using a TRIGGER to trigger a command running in Background, depending on the link below:
In my humble opinion: I think this kind of thing should be done at the application layer.