Can a black hole be formed by electromagnetic radiation?

The answer to your first question is yes.

Building on Demetrios Christodoulou's seminal work showing that black holes can form "generically" from focusing of gravitational waves starting from an initial space-time that is arbitrarily close to flat, Pin Yu has recently shown that one can also dynamically (and generically, in the sense that the formation is stable under small perturbations) form a black hole starting with only electromagnetic waves.

Of course, the interaction between electromagnetism and gravity means that as soon as you set the thing in motion, you will pick up gravitational radiation. And also that since a precise covariant notion of local gravitational energy is not available, the idea that the space-time starts out with only electromagnetic waves is a specific, frame dependent mathematical definition; one should keep that in mind before trying to draw too much physical significance out of the casual statement of the theorem.

For your specific second question, the answer is also yes. Einstein's equation specifies that $$ G_{\mu\nu} = T_{\mu\nu}$$ the left hand side, the Einstein tensor, is purely geometrical, and reflects the curvature of space-time. The right hand side comes from the energy-momentum contributions from the matter fields. The standard way of coupling electromagnetic waves to general relativity (Einstein-Maxwell theory) gives that the right hand side is zero only when the electromagnetic field vanishes. So the content of Einstein-Maxwell theory is based on that electromagnetic radiation can curve space-time.

In general relativity, the quantities that can act as sources of spacetime curvature are the ingredients of the energy momentum tensor. There is a nice diagram of these ingredients in the Wikipedia article.

Since radiation has such attributes (energy, momentum etc), it can act as a source for the gravitational field.

Edit: I seem to have omitted the very first part of your question. See the discussion here about whether or not there is a "critical energy density" which determines black hole formation.