Same as most modern audio players, MusicBee fully supports gapless playback.
And considering that (far as I know) nobody has reported such an issue before, and that foobar2000 also does not play your cd rips gapless, makes me think there is something else wrong. (assuming that the original CD does play the tracks gapless)
Perhaps a problematic cd-drive?
I'm not sure if it could be related, but have you set the correct offset value for it?