Short answer? Yes. Even though the James Web Space Telescope (JWST) is an infrared telescope, it's main camera, the NIRCam will effectively be able to take visible-light images of things between 7 and 13 billion years old.
For those unfamiliar, here is a diagram of the spectrum visible to us humans compared with the NIRCam on the Webb. As you can see there's barely any overlap.
However, because of Hubble's Law and the effects of redshift, visible light gets redder when it's ancient. Light from stars and galaxies that are at least 7 billion years old gets stretched out enough that it starts to fall entirely within the Webb's range.
That shift continues as things get older and older, and we run out of sensitivity once whatever we're looking at is about 13billion years old.
Curious as to how this all works? Luckily we just need a few points of cosmological groundwork to tie it all together.
First, if you look at a star 5 light years away from earth, you're actually seeing that star as it was 5 years ago because the light took 5 years to get to you.
Second, the universe is expanding, everything is always getting further away from everything else. This means that on very large scales the further away you are from something the faster it is moving away from you. Roughly speaking this is Hubble's Law. This together with the previous point means that the age of the light that you see from a star, it's speed away from you, and it's age are all tied together.
Finally, we need to understand redshift, which is probably best done by analogy.
So you know how when a train is rushing past you and it honks it sounds like "nnneeeyyaouwwwwwww" instead of the simple "nnaaaaaaaaaaaaa" you'd hear when it's standin still? That's because the sound waves get squished when it's coming at you and stretched out when it's rolling away. We hear the pitch get higher when the waves are squished, and lower when they're stretched. The same thing happens with light, but we percieve that change in "pitch" as a shift in color. You can think of blue light as being high pitched, and red as low pitched. As something moves faster and faster away from you, the "pitch" of it's light gets lower and lower and so the color gets redder and redder. This is called redshift. The opposite happens if something is moving toward you resulting in a blueshift.
So to bring this back around, what Hubble's Law is actually saying is that the older something it is, the more redshifted it will appear. With a little math we can see that visible light from beyond a certain age is redshifted enough to be visible to the JWST's infrared seeing eyes.