How to check peppers for bugs

Does it need checking? In principle no; there is only a concern of infestation. Remove the stem cap, soak, and wash inside and out. (Rav Moshe Vaya)

Rav Vaya’s rating

USA (bell pepper)	[★]	rarely infested, or after long storage
USA (fresh pepper)	✓	not commonly infested (muchzak k’naki)
Israel (bell pepper)	★ ☆	infested outside and inside
Israel (fresh pepper)	★ ☆	infested outside and inside

Rav Moshe Vaya, Bedikat HaMazon K’Halacha, Degree of Infestation chart, p. 936 (Red Sweetpepper) and p. 939 (Capsicum)

Which bugs

Thrips inside the pepper: young thrips are yellow, adult thrips are black.
As a general rule, any fruit or vegetable grown in a field can have insects clinging to the outside, so wash all produce well before eating. (Rav Moshe Vaya)

How to check

Cut the pepper in half lengthwise, or cut around the stem cap and remove it.
Soak in water with a little dish soap, about 3 minutes.
Wash well, inside and outside. (Rav Moshe Vaya)

Pictures

Red bell pepper. Photo: AntanO, CC BY-SA 4.0, via Wikimedia Commons.

Thrips (Frankliniella occidentalis), the kind of insect found in peppers. Scale bar: 1 mm. Photo: Jesse Rorabaugh, CC0, via Wikimedia Commons.

Sources

Rav Moshe Vaya, Bedikat HaMazon K’Halacha: summary sheet “Washing Fruits and Vegetables” (published by Machon Aliya); his written reply on the same page: a pepper needs no checking in principle, there is only a concern of infestation. Degree of Infestation chart, pp. 936 and 939.
cRc: remove the crown and stem area of every bell pepper, mini peppers too, and rinse and rub the outside under running water, especially in the grooves. Do not roast them whole. Hot peppers: a general look for obvious infestation. (Fruits & Vegetables Checking Guide)

References: cRc — Fruits & Vegetables Checking Guide · Rav Moshe Vaya — “Washing Fruits and Vegetables” (Machon Aliya, Hebrew)

Source: Rav Moshe Vaya, Bedikat HaMazon K’Halacha, pp. 936 and 939