If the speakers vibrate especially when the mega bass is on, does that mean that the speakers are blown on my icf-cdk50?

Download its user manual from here and refer it for details.

Look in the user guide...

http://www.retrevo.com/support/Sony-ICF-CDK50-Clock-Radios-manual/id/21951ci654/t/2/

It has NO outputs whatsoever but it DOES accept the audio from another device's headphone jack through its own stereo miniplug.

A common reason for 'distortion' with all bass amps can be either that the screws holding the speaker unit to the cabinet have worked loose or that there has been a build up of dust particles in the corrugations around the edge of the speaker cone, which rattle especially on the lower notes.
The solution is simple - remove the front grille and check the screws are tight. Use a vacuum cleaner to remove dust from the rim of the speaker cone.

It's not clear how the speakers sound with a different amp or if other speakers sound tinny with this amp. What's the source you're listening to? If it's Phono that opens up a different can of worms.

Can you certify that the electronics is faultless in this scenario? More info on that piece may be helpful.

The woofers could both be blown but I doubt it. Tweeters usally die if a speaker is overloaded. You should be able to see or feel the woofers vibrating in the presence of bass.

Mains transformer output volts are 12V AC for Sony ICF-CDK50
Circuit location is T901 and part number is 143591722 .... TRANSFORMER, POWER (US, CND)
Hope these info will help you.
Thanks.

sorry, I can't shed much light on this problem, but it appears to have been fixed. We played around with setting the time, although it didn't need fixing and with the timer.

I'm guessing you mean one of the bass drivers (as standmount or floorstanding speakers don't have subs as such). Place your hand lightly against the driver when playing some music at normal volume, there should be some vibration (this increases the louder you play). If there's no vibration, you're probably right.

Contact Mission, who can send you a new driver, contact a Mission dealer to see how much it'll cost to go through the dealer network. I'd recommend going down the latter route if you are unsure of what to do yourself.

Just noticed you posted this in March, in which case, I hope you got it fixed (or are enjoying your new pair instead!).