You might be able to simply add the shop as a defendant to your present lawsuit. Note that if the tatoo artist is truly an independent contractor — just rents the space from the shop and is not controlled by the shop like an employee would — then you'd have no claim against the shop for this.
That would not succeed. In order to win a case for emotional distress, the acts of the defendant have to be pretty extreme; something well outside the normal bounds of human conduct. A shop refusing to give you receipts or tell who the owner is might be frustrating to a customer but that sort of thing happens frequently and is not the kind of extremely outrageous conduct that would give rise to an emotional distress claim. As a fairly recent California court case explains:
A defendant's conduct is considered to be outrageous if "it is so ' " 'extreme as to exceed all bounds of that usually tolerated in a civilized community.' " ' " (Hughes v. Pair, supra, 46 Cal.4th at p. 1051, 95 Cal.Rptr.3d 636, 209 P.3d 963; see Potter v. Firestone Tire & Rubber Co. (1993) 6 Cal.4th 965, 1001, 25 Cal.Rptr.2d 550, 863 P.2d 795; see also Rest.2d Torts, § 46, com. d, p. 73.) Liability for IIED does not extend to " ' "mere insults, indignities, threats, annoyances, petty oppressions, or other trivialities." ' " (Hughes v. Pair, supra, 46 Cal.4th at p. 1051, 95 Cal.Rptr.3d 636, 209 P.3d 963.) Malicious or evil purpose is not essential to liability for IIED. (Spinks v. Equity Residential Briarwood Apartments (2009) 171 Cal.App.4th 1004, 1045, 90 Cal.Rptr.3d 453.
California's definition of extreme and outrageous conduct is based on comment d to section 46 of the Restatement Second of Torts. (See Hughes v. Pair, supra, 46 Cal.4th at p. 1051, 95 Cal.Rptr.3d 636, 209 P.3d 963.) Comment d to section 46 states: "Liability has been found only where the conduct has been so outrageous in character, and so extreme in degree, as to go beyond all possible bounds of decency, and to be regarded as atrocious, and utterly intolerable in a civilized community. Generally, the case is one in which the recitation of the facts to an average member of the community would arouse his resentment against the actor, and lead him to exclaim, 'Outrageous!' " (Rest.2d Torts, § 46, com. d, p. 73.)
Crouch v. Trinity Christian Ctr. of Santa Ana, Inc., 39 Cal. App. 5th 995, 1007, 253 Cal. Rptr. 3d 1, 13 (2019), review denied (Dec. 11, 2019). Note what the court says in particular in the second paragraph, that liability is only found "where the conduct has been so outrageous in character, and so extreme in degree, as to go beyond all possible bounds of decency, and to be regarded as atrocious, and utterly intolerable in a civilized community." Refusing to give you a receipt or tell you who owns the place simply doesn't rise to that level of outrageous conduct.
As for punitive damages, the shop refusing to give you a receipt or tell you who the owner is not the sort o thing that gives rise to punitive damages either as it's not wrongful conduct. The idea behind punitive damages is to punish and deter people from intentionally or knowingly doing things that are wrong or that they know will cause harm to others. Refusing to give a receipt or tell you who the owner was doesn't fall in that category. It was not wrongful to deny you those things nor was it something that the store would have any reason to believe would cause you great emotional harm. Most people would be at worst frustrated by that refusal, but not harmed. Again, from a California court case describing when punitive damages are warranted:
To contain the generosity and emotionality of juries, appellate courts have set the threshold high in defining situations in which punitive damages can be given. Conduct which may be characterized as unreasonable, negligent, grossly negligent or reckless does not satisfy the highly culpable state of mind warranting punitive damages. (G.D. Searle & Co., supra, 49 Cal.App.3d 22, 122 Cal.Rptr. 218.) Conduct which warrants punitive damages must be of "such severity or shocking character [as] warrants the same treatment as accorded to willful misconduct—conduct in which defendant intends to cause harm." (Nolin v. National Convenience Stores, Inc., supra, 95 Cal.App.3d 279, 157 Cal.Rptr. 32.)
Woolstrum v. Mailloux, 141 Cal. App. 3d Supp. 1, 10, 190 Cal. Rptr. 729, 735 (App. Dep't Super Ct. 1983).