Based on what I've read scrounging around the internet, it appears that the answer is no, Bottle and Cappy do not affect wasteland events.
From Section 20.9 (How do random quests with explorers work?) of therabidsquirel's The Fallout Shelter FAQ:
As explorers do their thing they can stumble upon random
quests...Discovering them is real time and you have to be in-game to
get the notification, so if you're only checking for short periods of
time your chances of getting a random quest are fairly low even with a
lot of explorers.
From Section 21.2 (What do they do when they visit?) of the same wiki, what Bottle and Cappy do affect is as follows:
Whenever they stop in a room there's a chance they'll start dancing.
If you tap on them while they're dancing you'll get a reward.
Cappy...rewards caps when tapped...Bottle...can reward caps or
Quantums. After a while...they'll leave.
For the duration of their visit they also provide a temporary +20%
happiness boost to everyone in the vault. If a dweller was higher than
80% happiness before the visit (including 100%) they will return to
exactly what they were before (unlike the small bonus from radio
The Fandom pages for Bottle and Cappy support the latter and neither make any mention of having any affect on wasteland events.
According to this Gamespot article, Bethesda released this statement regarding the Fallout Shelter 1.7 update:
To celebrate the launch of the Nuka-World add-on for Fallout 4, all
Fallout Shelter players will receive their first Special Event Quest:
a Nuka-World-themed mission which, once completed, allows Nuka-World
mascots Bottle and Cappy to appear in the Vault. Tapping on Bottle and
Cappy while they visit will reward players with Caps and Nuka-Cola
From Bethesda's statement, it would appear that Bottle and Cappy were only ever intended to produce a few extra caps and Nuka-Cola Quantum for the player.
This Eurogamer article for the same update says more or less the same, as does this Bethesda Support response and this review's screenshot caption.