I am currently getting my transcripts from University of Mumbai as well. I am getting them through http://mailtranscripts.com/. They are expensive if you compare them to getting the transcripts by yourself but since I didn’t want to inconvenience my parents, I took this approach. So far my experience has been alright. I feel they are a little slow and the communication is just okay. It might vary from agent to agent though.
I had to scan and email all the marksheet copies, an authorization letter, an application letter and an ID proof to MailTranscripts. They have submitted my application to the university and the ETA for receiving the transcripts is 15 days for the university and then some additional time for mailing. They have the option to send the documents directly to WES if you want. I have however asked them to deliver them to me since I need to add my US transcripts to the package before sending.
10 years ago when I got my transcripts from University of Mumbai, I had gone in person. I had to get transcripts from my college first and then take those along with the originals to some university office near Churchgate (I don’t know the exact address now but I think it was called Vidyapeeth Vidyarthi Bhavan and it was a stone’s throw from the Churchgate Station). It had its own bureaucracy; they had specific application drop-off times and a very short transcript pickup window back then. I don’t know if things have improved now.
If your cousin has the time for it, it might be better to get the transcripts in person otherwise you can check out MailTranscripts. Hope this helps!