Is it possible to replicate what SurveyMonkey has ? -  SMS Surveys | SurveyMonkey Enterprise
The goal is to have users take the survey via SMS, not just receive the invitation in SMS.

And I thought that SMS is dead already... 

I assume that with some custom coding and a SMS plataform that communicates with Limesurvey via the API you might be able to set something up like this. But I am not really sure if it is worth the effort.

SMS might really only work for very, very simple surveys. e.g. 1 or 2 closed responses. But what would be the advantage of SMS over a link sent to them? Today, most people will have a smartphone and can easily respond a couple of questions on the smartphone.

Unless you have a very, very specific use case or target group, I don't think it is worth investing much time and effort into this. But that is just my personal opinion. There might be use cases out there, where a survey over SMS is a totally valid use case, I currently just can't imagine one. 

Yes, SMS is still alive for some special scenarios.

Surveys for low-income participants who might not have access to data plans or Wi-Fi for example or a phone capable of using a web browser. It's another way to increase the saturation of your surveys.

Yeah, might be a little spoiled here in Brazil. Because if you don't have whatsapp, you basically "don't exist". Everything is via whatsapp, so that even the lowest SECs usually have some kind of internet capable phone with some prepaid data plan. Even up to high ages. This even has made communication easier for those that don't know how to read or write. They now send and receive voice messages and are back in the game.

Only SMS might only be relevant in very remote areas in the countryside here.

I am sure that you could built something with the api and platforms like trelo or something.

Sorry, it is not called trelo, but twilio. I am sure there are others out there as well. But of course this will require some custom coding.