Yes, she can. Lower-back tattoos (“tramp stamps”—sorry, ladies) are quite common these days, so the issue has been discussed over the past decade or so at meetings of the American Society of Anesthesiologists (ASA) and Society for Obstetric Anesthesia & Perinatology.
Lumbar epidurals require that a large-gauge needle be inserted through the skin (after numbing the spot with a small needle), so the concern is whether particles of tattoo ink (or the skin cells that contain them) might somehow be harmful if transported into the epidural space, or even the subarachnoid space which contains spinal fluid. (It is known that skin cells sometimes get deposited there, the needle being larger-gauge and blunter than most hypodermic needles.)
The concern is purely theoretical—there is no evidence of toxicity and no case reports of such a complication. The pigment particles are are considered biologically inert and are absorbed into skin cells called dermal macrophages (that’s what keeps the tattoo from disappearing.) The consensus—absent any large systematic studies—is that it’s safe.
The other concern is infection, which should be no riskier with a healed tattoo than with no tattoo. If the tattoo is fresh there might be increased risk. In any case no anesthetist would place an epidural through an area of infected or inflamed skin. So don’t get tattooed in your 3rd trimester!
My experience is that the L3–4 or L4–5 inter-spaces, where most epidural punctures are made, lie above the top margins of many such tattoos. Where the tattoo overlies the puncture site(s) there are often areas of the artwork with little or no ink. Since the skin stretches a little in any direction, the needle can usually be put through ink-free areas of skin without altering the anatomical target.
Bottom line: Don’t stress over it. Get the epidural if you seek pain-free labor. Discuss the issue with the anesthesiologist or nurse anesthetist who will perform the epidural.