I bought my PS4 with the Destiny bundle, but I lost the disc and can't play it. Is there a way I can still play it without the disc?
The only way to play a game without the CD is buying a digital version in the PS Store.
But actually you can find a new Destiny CD at a very low price (if you're talking about the first Destiny release, without any DLC).
There is no way for you to play a physical copy of Destiny without the disk. You only are able to play a game without a CD if you bought a digital copy of the game.
Short Answer: No.
Additionally, purchasing the Digital Download version from the PS Store will simply create a second installment on the PSHD, but your account itself is unaffected. The only thing you can do is re-purchase a physical copy but don't be disheartened, if you have DLC it will remain because it's tied to your user account and already unlocked and licensed to the account, not the physical media.
If you only had the physical disc, then you need that to play.
Best option would be to go to the PS Store (or Xbox Store) and buy a digital copy. It won't affect your character(s) as that info is stored on Bungie's servers.
A used Destiny disk will run you less than 10$. Put it in, and everything you've done up to this point will still be there.