What should I do if I can't get a copy of original transcript from the institute? I saw an offer for a Ph.D. position in US where they asked for original transcript which directly should be sent from the home institute. When I asked my institute, they said they don't have these kind of options. In this case what can I do? Should I write the programme co-ordinator?
Yes, you should write the coordinator at the program you're applying to, explain the situation and ask for permission to submit a copy of your transcript (I assume you have one?) rather than an original. To remove doubt that you are playing some funny games, offer to append a notarized statement from you attesting to the authenticity of the document, and say you'd be willing to take any other reasonable measures they would like you to take to accept the document.

Since the school is in the U.S., I'm guessing they'll be reasonable.