Was there still an orange cap on the needle of the syringe when Speaker Nancy Pelosi got vaccinated against Covid-19? No, that's not true: social media posts using that claim all use a photo taken at an angle that obscures the needle but at least one press photos taken at the same event clearly shows the needle.
An example of such a post (archived here) published on December 19, 2020 can be seen here. It paired a picture of Speaker Pelosi getting the vaccine and a zoomed in portion of the photo with following text:
Pelosi got the covid vaccine. Can someone explain how effective the vaccine is if you leave the cap on the needle You aren't fooling no one.
This is what the post looked like on Facebook at the time of writing:
(Source: Facebook screenshot taken on Sat Dec 19 19:30:58 2020 UTC)
The photo in question appears to be one of many pictures taken by the press pool on December 18, 2020 as Pelosi was getting the vaccine, an example of one of these photos can be seen in this PBS Newshour article. Speaker Pelosi herself also tweeted about the event:
Today, with confidence in science & at the direction of the Office of the Attending Physician, I received the COVID-19 vaccine. As the vaccine is being distributed, we must all continue mask wearing, social distancing & other science-based steps to save lives & crush the virus. pic.twitter.com/tijVCSnJd7-- Nancy Pelosi (@SpeakerPelosi) December 18, 2020
However because of the angle they were taken at in many of the photographs the needle wasn't actually visible.
But in at least one photograph taken by AP pool photographer Ken Cedino (archived here) the needle could be seen, it looks like it was used by the Daily Mail in this article: