Let's say I have a photon collector in orbit around the sun. It manages to collect photons perfectly efficiently, that is, without radiating off any energy.
Then, using Einstein's equation:
$$E = m c^2$$
since the collector is absorbing energy, its mass should increase.
Is this correct?
Yes.
I feel like there should be more of an explanation, but it's pretty straightforward. A blackbody absorbing energy will increase in mass. The absolute amount of increase is pretty miniscule, but it is not zero.
Since you ask about an object that does not also radiate energy, a blackhole might be a decent analogy. So, does a blackhole increase in mass when photons fall into it? Sure. If it helps, you can imagine that a photon of sufficiently high energy can produce pairs of electrons (or other particles) that could subsequently fall into the blackhole... well, except the antiparticle that would be annihilated shortly after creation. Either way, it seems easier to imagine the scenario with particles that have a rest mass because it more closely corresponds to our quotidian experience.
While it is true that the mass will increase your requirement of perfect absorption without radiation cannot be satisfied so easily. Total absorption would require a black surface at 0 K. Every surface would, according to the Planck's law and the Stefan Boltzmann law radiate thermal energy with $\propto T^4$. This limits the temperature rise to the surface temperature of the sun.
With the thermal energy of $k_B/2\cdot T$ per degree of freedom you can try to estimate the increase in mass between the minimum starting temperature and 6000 K for a given set of properties of the collector.
