"Can acupuncture help manage chronic pain, one of the signs of chronic Lyme Disease?" is an interesting question — not least because it sees two controversies unite. While chronic Lyme Disease isn't officially a thing at all , some papers describe acupuncture as no more than a "theatrical placebo" .
In the eyes of some, then, people hoping to manage chronic Lyme Disease with acupuncture are being treated for a condition that doesn't exist with a method that doesn't work.
Is it really that simple?
Chronic Lyme Disease, Pain, And Desperation: When Modern Medicine Doesn't Have The Answers
The authors of the paper that condemned acupuncture as useless were certainly right when they wrote, in their introduction, that:
"Pain is a big problem. If you read about pain management centers, you might think it had been solved. It has not. And when no effective treatment exists for a medical problem, it leads to a tendency to clutch at straws." 
They go on to describe acupuncture as such a straw, but in our context, there's a whole other straw — "chronic Lyme Disease" itself.
The non-existent diagnostic category of chronic Lyme Disease may in some cases be used to replace the preferred term of post-treatment Lyme Disease syndrome, in which people who have already had a Lyme Disease diagnosis and were treated for Lyme, with antibiotics, still have joy-squashing symptoms well after their treatment was completed. 
As one study noted, however, people also often end up with this dubious diagnosis without any clinical evidence they were ever even infected with the bacterium that causes Lyme Disease proper, Borrelia burgorferi. Whether you DIY this label or a practictioner of alternative medicine slaps it on you, it's clearly the result of looking for answers modern medicine simply hasn't been able to provide thus far. "Many patients referred for Lyme disease are ultimately found to have a rheumatologic or neurologic diagnosis", the study goes on to report, and "medically unexplained symptoms, whether resulting in entities such as fibromyalgia and chronic fatigue syndrome or syndromes with a less distinct pattern, account for most of the remaining patients who are diagnosed with CLD."  The study's authors further acknowledge that these latter, certainly debilitating, conditions "generally lack objective clinical or other objective abnormalities".
Can Acupuncture Help Manage Pain?
The team of people on the opposite side of the paper describing acupuncture as mere theatrical placebo proved equally competent at throwing verbal punches when they retorted:
"It seems somewhat naive to totally condemn the practice of acupuncture, while accepting orthodox medicine as the basis for treating all medical conditions." 
If modern medicine proved ineffective at managing the pain you suffer from as the result of either post-treatment Lyme Disease syndrome or an as of yet unclear condition you think is chronic Lyme Disease after repeatedly looking to its practitioners for help, you already know that modern medicine has failed you so far.
From research that suggests other mammals — unaffected by the placebo effect — react to acupuncture similarly as humans do  to studies that indicate that "acupuncture has a clincally relevant effect on chronic pain that persists over time" , the currently available data makes it clear that acupuncture isn't in the same ballpark as, say, Shamanic healing. There's a reason the World Health Organization says acupuncture can be helpful in the management of over 40 different conditions, including chronic pain. With the addition that "acupuncture treatment is [generally] safe if it is performed properly by a well-trained practitioner" , it seems that you have little to lose in trying acupuncture for chronic Lyme Disease.
Pain is, ultimately, subjective. If acupuncture causes you to experience less pain without leading to side effects, that's not something anyone can really argue with.