The party’s national executive committee (NEC) unanimously agreed the decision on Saturday – on the recommendation of leader Nicola Sturgeon.
The SNP’s ruling body said the party has the “legal standing” to put forward its political case.
Lawyers are now being instructed to prepare the application and it is expected to be filed with the court in due course.
It comes as Lord Advocate Dorothy Bain QC referred a prospective referendum Bill to the UK Supreme Court to determine if it is within Holyrood’s powers.
The latest court document filed by Scotland’s top law officer said on Friday that the case for holding an independence referendum leaned heavily on the vote being “advisory”.
In a 51-page filing, the Lord Advocate argued the Bill would not be “self-executing” and would have no real effect on the union – which is reserved to Westminster – but would simply be to “ascertain the wishes of the people of Scotland on their future”.
The arguments are expected to be heard by the court on October 11 and 12.
Kirsten Oswald, the SNP’s business convener, welcomed the reference to the Supreme Court.
She said: “It is important that there is clarity as soon as possible on whether or not a referendum without a Section 30 order is within the competence of the Scottish Parliament.
“That clarity will, we hope, pave the way for a referendum on October 19 next year, and this is what the SNP is actively planning for.
“Given the importance of this case to the future of Scotland, the NEC has decided that it is appropriate for the party to make an application to intervene.
“The NEC wishes to give full support to the argument that an advisory referendum is within the power of the Scottish Parliament, and also make the case that Scotland has a right to self- determination, which – as things stand with Westminster’s democracy denial – can only be upheld through legislation in the Scottish Parliament under its current powers.”
The SNP will be able to put forward its political case if the application is approved.
Ms Oswald continued: “As the largest independence-supporting party – which has won every election in Scotland in the past decade – we consider that the SNP has legal standing to represent the interests of the very significant proportion of the population who support Scotland becoming an independent country.
“We also have an important perspective on the legal issues that the court will consider and, in particular, the importance of these matters being decided in a way that upholds the right of the people of Scotland to have their say and express their democratic will.
“Whether or not to grant the application is, of course, a matter for the court.”