Grant permissions to run an SQL server job
You could create a stored procedure that runs the job.  You can use with execute as owner to run the stored procedure as the database owner.  That way, the users themselves don't need permissons on sp_start_job.

create procedure dbo.DoYourJob
with execute as owner
as
exec sp_start_job @job_name = 'YourJob'


Grant execute rights on DoYourJob to allow people to start the job.

Of course, you can also put the contents of the job in a procedure, and grant rights to run that.  That would allow more interaction, like displaying a result text.