I'm sure the audio guys will chime in here with more knowledge than I.
Back when I was doing radio commercials and imaging, I used quite a few sound effects libraries - we had the entire Sony Pictures SFX collection, the BBC SFX collection, The Hollywood Edge SFX collection and another collection the name of which escapes me. There were anywhere from 10 to 50+ CDs in each collection and each came with a large booklet that was an alphabetised directory of the sound effects included.
We then had what I called 'production SFX' which were zips and zaps and buzzes and booms and bangs and blips and warped FX that were all the little stingers and effects that weren't necessarily 'real world' FX. Killer Hertz was one of the better suppliers of that kinda thing, but there were heaps of suppliers around. We used to get new CDs of these 'production FX' every month
For Production Music, we would get sent CDs from each production house every month, the most common being Beatbox, Universal, KPM/EMI and then Extreme Music. Every month we'd have to send in a written report of what tracks we'd used from what companies wihch often ended up being hundreds of tracks per month (I always made my assistant do the paperwork
)
Needless to say, the walls and cupboards of our control rooms were lined with CDs..
These days, I do keep an account with Extreme Music, and I do have a few SFX collections ripped to my computer for the odd job that comes up here and there (especially good for corporates!)
In radio, we simply didn't have the time to record all our own sounds, nor would it at all be feasible to bring in and record different sound effects, especially with the vocal booths being taken up by shows, news, or V/O records for different segments or different commercials or promos.
I know a lot of audio post guys like to record a lot of their own sounds, often using Foley artists, and of course music is generally provided by a composer.
However, I quite often hear on television promos and commercials, as well as current affair stories, music beds that I've used for any number of radio promos. The music from Extreme is used on heaps of things, and Two Steps From Hell who now work with Extreme have been sort of a 'go-to' in terms of trailer music.
In terms of SFX, I imagine major audio post facilities keep a library of their own sounds that have been recorded over the years, as well as the SFX libraries available on CD.