We're a 50+ couple flying to India this summer. I'm trying to buy domestic flight tickets from Delhi to Kullu, from Jaipur to Delhi and from Amritsar to Rishikesh. When I tried to buy the tickets on the Air India website, the purchase couldn't be processed although I tried numerous times. If it might be relevant, I'm from Israel. Is there an alternative website for purchasing these tickets?
There are multiple websites that you can use for domestic flight tickets in India.
Apart from the ones mentioned in the answers. I suggest you look at https://www.skyscanner.co.in/ to get the consolidated results. There you can also find some websites that accept the payment in dollars or your currency (they run under the Indian domain, however, the operations are managed overseas) and at times offer the cheapest rates.
There are multiple websites that aggregate the airlines fares and allow booking. 2 are already mentioned above, apart from this there are more:
Note: Some of these sites may have restrictions on using an Indian credit card. If you face this issue, best call the airlines office directly and book.
Apart from Air India there are other airlines which have great service: Jet Airways, Air Vistara, Indigo