Einstein's equivalence principle says that gravitational and inertial acceleration are locally indistinguishable from each other.
Correct. Constant acceleration is completely indistinguishable from a perfectly uniform gravitational field, but no real gravitational field is perfectly uniform, so there are tidal effects if you make measurements over a large enough distance. When we say "locally indistinguishable" it means you confine your measurements to a chunk of spacetime that's small enough in the given gravitational field so that it looks like flat spacetime, and then the tidal effects are negligible.
what happens to a craft that is accelerated indefinitely
If you apply uniform acceleration to a craft, that is, the acceleration feels like constant gravity to the people in the craft, then its speed gets closer and closer to c without ever reaching it. That may seem weird, but it works that way because of the relativistic formula for velocity addition. For further details, please see the classic Usenet Relativistic Rocket page.
I assume that this constant input of energy would eventually cause the object to collapse into a black hole! Is that true?
No, that's not true. Even if the speed of the craft is a tiny fraction just under lightspeed, so the crew are experiencing a huge time dilation relative to the rest frame they took off from, they won't notice anything unusual, although their view of the universe will change due to length contraction and Doppler shift.
However, if such an ultra-relativistic craft crashed into something, that could create a black hole, but the required kinetic energy is incredibly high. Eg, if you could turn all the energy of a dozen Earths colliding with a dozen antimatter Earths into kinetic energy, and used that energy to slam two billiard balls into each other, it could create a black hole, providing the collision fragments stay close enough to the collision point.