iTunes will add the artwork to the file itself, with two exceptions, one "glaring" while the other is not so obvious.
Although you refer to MP3 files, there are several file types that you may be using, for various reasons. They all work the same way. In your iTunes Library, when you select an album of songs and use Edit/Get Info/Artwork, and either paste artwork into the artwork panel, or use Add Artwork (and then go to the storage location of the artwork), iTunes adds the picture to each song file. So if you then play the song in another programme, you will see the artwork. If you move the file, the artwork will go with it. This works for almost every audio file type.
The glaring exception is .wav files. Artwork cannot be added to wav files, so if you "add artwork" in iTunes to a wav file, that is the time that iTunes keeps the picture somewhere else and refers to that when it needs to display it. Consequently, if you copy that wav file to somewhere else, the artwork will not go with it.
The less obvious issue is if you use artwork that does not conform the the standard requirements, such as images that are too large. I usually use images that are 300x300, or as lagre as 600x600 and in jpg format.