My character recently obtained a vorpal weapon. I was wondering whether I could enchant it with an effect similar to the Surge of Fortune spell (from D&D 3.5 edition).
In 5th Edition, there are a couple of different things that reference the face value on the die, regardless of modifiers. This is commonly referred to as a "natural roll" - whatever the physical die actually shows - but "natural" it isn't actually a term used in the game rules in this way.
Note that the Vorpal Sword's special effect explicitly triggers on roll of 20, not a critical hit. Things that might increase your critical chance do not affect this - a Champion's Improved Critical attack rolls are no more likely to trigger a vorpal weapon than anybody else's attack rolls.
Your best bet is to make lots of attacks and find ways to roll with advantage. Things that let you reroll dice, like the Lucky feat or Elven Accuracy if the Vorpal Sword is a scimitar, can be beneficial, too.
Another option, though not especially probable, is to have access to a friendly Diviner. If one of the diviner's Portent dice comes up 20, he would be wise to hold onto it to apply to the character swinging the Vorpal Sword.
There is no reliable means for auto rolling a natural 20
In 5e there is no mechanic for auto-ROLLING a 20. There is a mechanic for automatic criticals (see below), but that isn't as meaningful with the Vorpal Sword requirements.
However, if you happen to have a Divination Wizard with you who also happened to roll a 20 for the Portent roll, you would be able to use that. But that's a lot of IFs (PHB, 116).
Increase your odds with Lucky and Advantage
Findings ways to get additional rolls, via the Lucky Feat or through Advantage to increase your chances of rolling that Natural 20 are still an option.
Unrelated to Vorpal: But Conditions are your source for Auto-Crits
There isn't anything to buff YOU to get an auto crit, but you can debuff an enemy. Anything that can gives the conditions Paralyzed or Unconscious provides the following effect (PHB, 291 and 292 respectively):
Any attack that hits the creature is a critical hit if the attacker is within 5 feet of the creature.
NOTE: You must be within 5' for this to trigger
No, there is only advantage
There are multiple ways to get advantage, which allows you to roll two times, which in turn makes it a lot easier to roll a natural twenty. But you still need to roll the 20 - the vorpal sword is not about getting a critical, but about rolling a twenty.
If it were about a critical you could for example look at Rogues with the Assassin Archetype, or Barbarians that get a critical on 19, or some other features - but these are not what triggers the vorpal sword effect.
Apart from the Wizard's Portent class feature, which allows you to roll up to three times at the start of the day and later use these rolls, you can only try to use advantage, for example by being stealthy or circumstances or spells that explicitly say so...
There is only one official feature that lets you manipulate what you roll: the Wizard School of Divination's Portent (PHB 116). And even then it is not guaranteed, you have to roll a 20 out of 2 (3 on higher levels) d20s after a long rest.