Not in Utah, see SB 3007, which confers immunity from liability arising from covid except for willful misconduct, reckless infliction of harm or
intentional infliction of harm. No such law has been passed in Washington state: so it depends in part on whether an immunity law was enacted (Georgia, Louisiana, North Carolina, Oklahoma and Wyoming have). Immunity may not extend to churches in some states (which would probably cause a First Amendment lawsuit, if put to the test). A federal law was proposed which would have included immunity to tort liability for churches, but it didn't go anywhere.
Since the question stipulates that plaintiff can prove that "the source of the illness was one of these religious gatherings", we needn't discuss the hardest part of such a suit. Negligently transmitting a disease has long been recognized as a cause of action (Bilo v. Allegheny Steel). Also, a person can be liable both is they have actual knowledge of the disease or constructive knowledge (they should know from the circumstances), see John B. v. Superior Court. A complete lack of knowledge of the harm would ordinarily be a barrier to liability, but it remains completely undetermined whether everybody is or will be legally deemed to have constructive knowledge that they have covid. If a person has actual knowledge that they have the disease, or have clear symptoms (traditional constructive knowledge), then they could be held liable for transmitting the disease. If they have no clue that they have it yet they do transmit the disease (this seems to happen), then under normal principles of liability, the person is not liable – unless the courts start reasoning that everyone ought to "know" that they have the disease.
The legal requirements imposed on entities and individuals in the US is extremely variable and ever-changing: at any given time, there is some prohibition. If businesses are allowed to operate at 25% capacity and only allowed to serve people wearing a particular kind of mask, they can be penalized for operating at a higher capacity or not enforcing the mask rules. If there are no restrictions on a business, they cannot be penalized for not obeying a restriction that exists in a different jurisdiction, or a restriction that can be imagined but was not enacted, or a restriction that was cancelled. Choosing to not impose the most stringent imaginable restrictions (complete shutdown and quarantine, do not go outside) does not create liability for negligent action, because negligence is based on the distinction between reasonable vs. unreasonable actions, and at least so far, operating a business or church that people can go to (and catch a disease) is not currently held to be an absolutely unreasonably incautious act. However, in some states, violating health regulations does create per se negligence, so if a church ignores the legal restrictions, they they could be in line for liability – but that may only apply to legislative mandates and not executive mandates (as I believe all lockdown orders in the US are).
The person contracting the disease is likely to be found to be contributorily negligent by the same reasoning that assigns blame to the church / church-goer, because they should know that going to church increases their risk of catching the disease. In some states (like North Carolina), that means that there is little chance to get anything from whoever gave them the disease. In most states, the question would be, who is more at fault, the person who "delivered" the disease or the person who went out and got the disease? That could adjust the liability of the plaintiff to the point that they just wasted their time and money with the lawsuit.
The courts would have to determine, did the plaintiff do enough to protect themselves from the disease (after all, they did not stay home)? Did the individual defendant do enough to prevent their transmission of the disease (did they knowingly go to church with the disease or were they unaware; did they have a suitable mask or were they unmasked)? What this comes down to is, did all parties do enough to prevent the spread of the disease?