Offensive rating increases from 1 to 4.
First of all, you have a +3 sword. That means +3 attack bonus and +3 damage. Replacing the morningstar with a Vorpal longsword carries the attack bonus from +4 to +7 and the damage from 11 to 14.
Mathematically, the beheading is worth at most two points of damage. When considering tier one PCs, they usually won't have a lot more than 40 hit points (at most 49 for a 4th level barbarian with Con +4). ButThe average tier one character (if there is such a thing) probably has less, but let's assume that the odd crit kills a healthy 40 hit point character. This is worth 40 damage, or 2 points of damage per attack since the odds are one in twenty. Multiattack and advantage would change the odds but the bugbear does not have those.
So, the final damage output would be 16 (CR 2) increased to CR 4 (attack bonus +7 instead of +3).
By the book, the final CR would still be 1 (4 and 1/4) but that is because at CR below one, the monster creation table allows for many more hit points than the MM monsters usually have at that CR.
The monster is skewed and the standard deviation problematic
The offensive CR is much higher than the defensive CR. This means that the monster will easily kill someone if the initiative and attack / damage rolls go right (wrong from the players' perspective). This, however, is mostly due to the +3 and not the beheading.
The other problem is the standard deviation (your monster is now very swingy). Every natural 20 will kill a PC. Probably this will not come up: The chances are slim and the monster can even be restrained / disarmed etc. as NathanS stated in their answer. If it does happen, somebody dies. Now, the random critical hit killing a poor first level character is nothing new. However, the Vorpal sword significantly aggravates the problem because:
a critical hit does not often kill a level 1 character, but always
the principle got extended to later levels
You cannot reasonably consider that large a deviation in CR. If you increase the CR (the number, not what it means) and no one gets beheaded, the monster will seem underwhelming. If you take a low CR and someone dies it will seem like the monster was too dangerous. However, if the players know the dangers and can take countermeasures, it doesn't need to come to that, see NathanS's answer. The principle is the same as with a Will-o'-wisp, which will also kill a PC if you let them lie around unconscious.