Removing /home LVM and merging in to /

Yes when you do the lvremove (warning: this kills the data) on the vg_vpsny23-lv_home volume, the space will become available in the volume group again which will let you do a lvextend on the vg_vpsny23-lv_root volume. In other words:

# lvremove /dev/mapper/vg_vpsny23-lv_home
# lvextend -l +100%FREE -r /dev/mapper/vg_vpsny23-lv_root

This should extend the root volume online. Remember that you can grow a filesystem online but you have to unmount a filesystem to shrink it.

For the root filesystem, taking it offline means booting into rescue mode. So if you may want to use some of this space elsewhere you may want to modify the argument to the -l option that I gave you up there.

Do not forget to comment out or remove the line from /etc/fstab that refers to /home. If you leave this in then Linux gets confused shutting down and restarting.

Also worth noting that if you receive the message:

Logical volume vg_somedisk/lv_home contains a filesystem in use.

do a umount -a to un-mount the volume, perform the resize, edit /ect/fstab then a mount -a.