The passage you quoted in your question is from Shreemad Bhagavata Maha Purana 4.4.17 and it says that "one should by force cut out the blasphemer's tongue and kill the offender". And your enquiry is about "does Vaishnavism ordains one to the cut-the tongue of a blasphemer? "
The answer is No. Not only Vaishnavism but none of the sects of Hinduism advocate or endorse such an act, nor such a thing is said anywhere in Hinduism scriptures.
Your quote is a classical case of what is called cherry-picking the saying of gods and applying that in the human context. In this case, this sentence is said by Sati the wife of Supreme Lord Shiva himself, Sati is a goddess. Now let's see in which real context the above passage is narrated by Vyasa in the Bhagavata Purana.
The Chapter 4.4.17 is about Daksha yajna and goddess Sati's self-immolation in that when Daksha here father and other gods insulted lord Shiva at that time.
Daksha Prajapati the father of Sati and father in law of Lord Shiva was not good at Lord Shiva.
Then goddess sati in anger spoke to Daksha her father and said the above. Note that she was addressing the Daksha and the gods present there and not to Vaishnava devotees nor to us humans. Seeing that her husband is insulted she advocated lord Shivas auspiciousness.
So we have to look at the passage from the context of the storyline and not in the general sense. Sati was furious at that time because her husband Shiva was insulted in front of all the gods, She is not ordering or asking to do this act either Lord Visnu's or Lord Shiva's devotees, but just in anger in order to convey that insulting lord Shiva in such a way is great sin and the sinner should be punished in such away (by a powerful man or god) only i.e. who do have the authority to do so and not by all. She is also saying a devotee if unable to do so should lay down his own life in honour of Lord Shiva.
So all this passage should be understood from the point of view of Daksha yajna and goddess Sati's self-immolation. And not to take the word in the literal sense, this is not the order of gods, but just the angry words of goddess Sati as she was in a rage.
Hinduism or Sanatan Dharma is the most tolerant religion and no sect follow such thing. We can also see that many times god himself took avatar to kill sinful humans that does not mean that Hinduism ordains such act.