can any govt agency tap mobile phones

I think the police, on proper ground, can.

That is illegal if not approved by  from Govt. under Telecrapich Act;

17[5. Power for Government to take possession of licensed telegraphs and to order interception of messages.—(1) On the occurrence of any public emergency, or in the interest of the public safety, the Central Government or a State Government or any officer specially authorized in this behalf by the Central Government or a State Government may, if satisfied that it is necessary or expedient so to do, take temporary possession (for so long as the public emergency exists or the interest of the public safety requires the taking of such action) of any telegraph established, maintained or worked by any person licensed under this Act. (2) On the occurrence of any public emergency, or in the interest of the public safety, the Central Government or a State Government or any officer specially authorised in this behalf by the Central Government or a State Government may, if satisfied that it is necessary or expedient so to do in the interests of the sovereignty and integrity of India, the security of the State, friendly relations with foreign States or public order or for preventing incitement to the commission of an offence, for reasons to be recorded in writing, by order, direct that any message or class of messages to or from any person or class of persons, or relating to any particular subject, brought for transmission by or transmitted or received by any telegraph, shall not be transmitted, or shall be intercepted or 13 Substituted by Act 45 of 1998, section 3 for ‘the Provinces’. 14 Substituted by Act 45 of 1998, section 3 for ‘the Provinces’. 15 Inserted by the Indian Telegraph (Amendment) Act, 2003. Act No. 8 of 2004, received the assent of the President on the 9 January, 2004, with retrospective effect from 1.4.2002. 16 Inserted by the Act 7 of 1914. 17 Substituted by the Indian Telegraph (Amendment) Act, 1972, with effect from 21.8.1972. 38 detained, or shall be disclosed to the Government making the order or an officer thereof mentioned in the order: Provided that press messages intended to be published in India of correspondents accredited to the Central Government or a State Government shall not be intercepted or detained, un

and in addition, issue a notice and get information (RTI or CPACT) as to the entire correspondence and certified copies of orders on such interception of calls from a request from those authorities to final orders by a competent authority of service provider and such dates with timings.