The mount command requires superuser privileges, but you could mimic the effect of running a command without sudo by allowing user to run the command without a password, then adding an alias that invokes the sudo for you. For example:
Add this in
$user with the username (you might want the umount command as well):
$user ALL = NOPASSWD: /usr/bin/mount, /usr/bin/umount
Add this to your shell rc, e.g.
alias mount="sudo mount"
alias umount="sudo umount"
Apply to the current shell session:
Edit: the shell aliases should be one per line (as it did appear in preview), not sure why they got strung together after posting.
Another option is to use some utility like udisksctl which is already running as root to mount, e.g.
udisksctl mount -b /dev/sdbN (mounting to