Pfizer's and even more obscure mRNA vaccines that haven't been approved have an INN, which ends with -meran (e.g. CureVac's is "zorecimeran"). But as far as I can tell Moderna's vaccine doesn't have an INN. Why is that? Do they need to apply to the WHO for an INN (and they haven't done that)?
The "-meran" suffix was selected by the World Health Organisation (WHO) as a common stem suffix for mRNA drug products in 2016 (though the technology originated in the early 1990s). Curevac were involved in pioneering this technology and they made the application. See this article announcing the news.
From the WHO guidance on International Nonproprietary Name (INN):
Another important feature of the INN system is that the names of pharmacologically-related substances demonstrate their relationship by using a common "stem". By the use of common stems the medical practitioner, the pharmacist, or anyone dealing with pharmaceutical products can recognize that the substance belongs to a group of substances having similar pharmacological activity.
The extent of INN utilization is expanding with the increase in the number of names. Its wide application and global recognition are also due to close collaboration in the process of INN selection with numerous national drug nomenclature bodies. The increasing coverage of the drug-name area by INN has led to the situation whereby the majority of pharmaceutical substances used today in medical practice are designated by an INN. The use of INN is already common in research and clinical documentation, while their importance is growing further due to expanding use of generic names for pharmaceutical products.
While INN use is widespread, it is not complete, nor is it a requirement for drug approval at local or national level.
The process of INN selection follows three main steps:
A request/application is made by the manufacturer or inventor;
After a review of the request a proposed INN is selected and published for comments;
After a time-period for objections has lapsed, the name will obtain the status of a recommended INN and will be published as such if no objection has been raised.
The process is entirely separate from drug approval through national agencies (like the FDA in the USA, EMA in the EU and MHRA in the UK). It may take some time to complete as it includes time for objections to be registered. Commonly, drugs are referred to by a code name during trials, before being assigned an INN and of course a proprietary name later.
For example, the Pfizer-BioNTech COVID-19 vaccine has an INN of "Tozinameran" but was code-named "BNT162b2" during development and trials.
Specific guidance is available on the application process for nucleus acid drugs (like mRNA).
INN are selected in principle only for single, well-defined substances that can be unequivocally characterized by a chemical name (or formula). It is the policy of the INN programme not to select names for mixtures of substances, while substances that are not fully characterized are included in the INN system in exceptional cases only.
Not every substance can or should have an INN. However, as you rightly state, mRNA vaccines are getting INNs ending in the "-meran" suffix.
In summary, I suspect that the Moderna vaccine will get a INN with a "-meran" suffix, but as these vaccines are only just being approved for emergency use, the process has not completed yet. I could not find any information on the status (if any) of an INN application for Moderna’s mRNA-1273 vaccine, but I did find an update from the EMA about the approval process.