Yes, the bard gets 1/2 proficiency to initiative.
In 5e all checks are ability checks. This is why every check in published materials is listed as Ability (Name), such as Strength (Athletics).
If you can add an ability modifier to a roll, it's an ability check (unless it's a saving throw or an attack, those aren't ability checks). If you happen to have a skill related to that ability, and you are proficient in that skill, you can add your proficiency bonus.
Intiative is a Dex check with no potential skill attached to it. So things that modify Dex checks work. Other things modify your ability checks as well (such as adding all or half of your proficiency bonus. The bard has one example of this.
The Champion Fighter's L7 feature has another:
Starting at 7th level, you can add half your proficiency bonus (round up) to any Strength, Dexterity, or Constitution check you make that doesn’t already use your proficiency bonus. (BD&D 26)
Since Initiative is an ability check that uses your Dex, the Champion Fighter also gets half their proficiency to initiative.