Disclaimer: The following answer has been written from a Ramanandi Vaishnava perspective.
Ramanandi Vaishnavas have absolutely no problem in accepting the fact that Lord Shiva is the deity being extolled in the Mahamrityunjaya Mantra since Jagadguru Ramanandacharya and great Ramanandi saints such as Tulsidas has accepted Hari-Hara-abheda and that Shiva, like Vishnu, is indeed a bestower of moksha.
However, Ramanandi Vaishnavas also believe that the Mahamrityunjaya Mantra can also be interpreted to be referring to Sri Rama Himself. This is how Swami Ramabhadracharya interprets the Mahamrityunjaya Mantra in one of his lectures:
Vashishthaji is its Rishi. Meaning Vashishthaji is the one who recieved this mantra...He said "Tryambakam Yajamahe". What did he say? "Tryambakam". What does "Tryambakam" mean? Generally, Vedic scholars will interpret it to be referring to the three-eyed Shiva. That is indeed the correct interpretation but it also has another interpretation. "Tryambakam Yajamahe". Three Ambas i.e. the one who has three mothers: Kaushalya, Kaikeyi and Sumitra. "Tryambak", what does "Amba" mean? Mother. So what does "Tryambakam" mean? The one who has three mothers, that person "yajamahe" i.e. Lord Sri Rama we worship..."Tryambakam Yajamahe"- we worship the one who has three mothers...sugandhim pustivardhanam. From who's body is the sacred fragrance being emanated. Pustivardhanam, who is healthy. What does "Pusti" mean? To nourish. The dependence of devotees over the Lord and the Lord's blessings over His devotees is what "Pusti" refers to. (Translated from Hindi)