This was prepared on the stated assumption that the performances would be unpaid. OP later stated We aren't going to get paid (if they pay us, it'll be a nice bonus!) but we do expect to be reimbursed for most or all of our travel costs.
Some Indian visa regulations on the web seem to be described about as clearly as a pea souper at night during a power cut for a blind person down a mine. However, there seems to be just about enough indication that a good place to start would be with a Group Application for an Employment visa.
Yes the Indian Embassy in Rome (as mentioned by @anshabhi) does state:
Those going to India for Charitable performances e.g an artist going for painting exhibition, musician for musical shows etc on charitable basis have to apply for Entry Visa.
This though may be a mistake or a mistranslation because more than one other site (eg immihelp for Entry visa) shows:
An Entry visa is granted to the following types of applicants:
• A person of Indian Origin, defined as a person who has held Indian nationality or is a child or grandchild of a person who has held Indian nationality.
In fact, persons of Indian Origin must apply for an Entry Visa instead of a Tourist Visa.
• A spouse or child of a person of Indian Origin.
• A spouse or dependent family member of a foreign national coming to India on a long term visa such as Employment, Business, Research or Student. In such cases, the duration of the visa granted will end on the same date as the visa duration of the principal visa holder.
Maybe it applies to the above and to others but I have seen no sign that that is so.
The Indian Embassy in Rome does mention performing artists and seems to differentiate charitable basis from remunerated performances, but that is little help when applicants are in USA, have to apply (pity them) through Cox & Kings and Cox & Kings seems barely to acknowledge the existence of performances on a charitable basis.
However, it does acknowledge some performers. If you have several days to spare poking around in that appalling website you might stumble across:
Eligibility 2. Foreign artists engaged to conduct regular performances for the duration of the employment contract given by Hotels, Clubs or other organizations.
No mention in the OP of "regular performances", "employment contract", "Hotel, Clubs or other organizations" but at least "Foreign artists" matches up.
On that basis, poke around for another couple of days and you might stumble across:
So (obviously¡) if for India you want a Volunteer visa, apply for an Employment visa. immihelp (which does seem indeed to be helpful) has:
A 'EMPLOYMENT VISA' is available for travelling to India for the purpose of working in India, for an organization registered in India. An Employment Visa is granted to an employee or paid intern of an Indian company and to persons traveling to India for volunteer work with a Non-Governmental Organization (NGO).
Now NGOs etc are an added complication but at least that is some reassurance that volunteering is deemed subject to an "Employment visa".
Next, "group application". Search that at The Embassy of Washington D.C., USA and at Cox & Kings and there are no hits. However such things are definitely possible, according to some, at least for "group tours":
There is also the somewhat special case for Entry visa (the one that seem applicable essentially only for family members of Indian origin):
(GOVERNMENT of INDIA). The site shows a button "Apply For Group/Family Members" (however the site seems may apply only to residents of Pakistan, though I don't see it bothering to mention that anywhere).
But it is quite possibly that though available to some categories a group application for Employment visa is not available. Unless up-to-date practical experience is forthcoming the best option might be to attempt to complete the necessary documentation and if at some point in that process it becomes clear that forms are only allowing entry of details for one individual at time, suspend entry and try contacting Cox & Kings (516-206-1483 / 646-589-0088 Hours of Operations: 9.00 AM to 9.00 PM EST (Contact Center operates from Monday to Friday except Holidays).