I know in some homebrew classes, such as Matt Mercer's Gunslinger Archetype, you can add your proficiency bonus to your initiative modifer. Are there any ways of doing this in the official rulebooks?
No. There are a number of abilities that come close, but there is currently no official method to add your proficiency bonus to the Dexterity check you make for initiative.
- The Barbarian gains advantage on initiative checks with Feral Instinct
- The Bard adds half their proficiency bonus to initiative with Jack of All Trades
- The Fighter (Champion) adds half their proficiency bonus to initiative with Remarkable Athlete
- The Rogue (Swashbuckler) adds their Charisma bonus to initiative
- The Ranger (Gloom Stalker) adds the Wisdom bonus to initiative
- The Enhance Ability spell can give advantage on initiative with the Cat's Grace option
- The Alert feat gives a +5 bonus to initiative (note that this is as good as or better than proficiency at all levels before 17)
- The Rod of Alertness gives advantage on initiative checks
- The Sentinel Shield gives advantage on initiative checks
- The Weapon of Warning gives advantage on initiative checks
In addition to all the game features in Miniman's answer, there's also methods of improving your improving your initiative that rely on the fact that initiative rolls count as ability checks:
- Inspiration can be used to gain advantage on an ability check of your choice.
- The Guidance cantrip, available to Druids and Clerics (and other classes through Magic Initiate or multi-classing.) The cantrip allows adding a 1d4 to an ability check of your choice.
- Bardic Inspiration allows adding a 1d6 to an ability check.
- Halfling's Lucky racial trait allows rerolling 1's on ability checks.
- The Lucky feat allows you to roll a second d20 whenever you make an ability check up to 3 times per long rest.
Rogue's Reliable Talent turns any d20 roll lower than 10 into a 10 on ability checks that let you add your proficiency bonus. Through multiclassing it's possible to have both Reliable Talent and Jack of All Trades or Remarkable Athletes, which let you add half your proficiency bonus to dexterity checks that don't already allow you to add your proficiency bonus. Adding half or double your proficiency bonus still counts as adding your proficiency bonus, as confirmed in an unofficial tweet by Jeremy Crawford:
For a rogue/fighter or rogue/bard, Reliable Talent does work with Remarkable Athlete and Jack of All Trades. #DnD
This is also implied by the Proficiency Bonus rules (PHB, p. 173), which state:
Your proficiency bonus can't be added to a single die roll or other number more than once. (...) Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.
The rules still use the term added even though they acknowledge the possibility that your proficiency bonus might be halved or doubled.