First note that being an EFTA member does not bring much in itself nowadays (only marginally easier trade with a handful of small countries). It used to be roughly equivalent in size to the EU (EC at the time) and thus position itself as an alternative at a times where external tariffs were much higher but nowadays trade with the EU is very open and the EFTA only includes Switzerland, Liechtenstein, Iceland and Norway… EFTA membership does not give a country access to the EU single/common/internal market.
The main reason the EFTA still exists is as a vehicle to coordinate with the EU and to manage the EEA, for that's where things are really happening. Case in point, the confusingly named “EFTA court” only includes EEA participants (and not Switzerland). And Switzerland, where the EEA agreement was rejected in a referendum, ended up concluding a series of bilateral agreements with the EU to get essentially the same thing.
In any case, being part of the Schengen area is not a requirement to be part of either the EFTA or EEA. Several cues to that can be found on the EFTA's website. For example the page about the EEA agreement:
The EEA Agreement does not cover the following EU policies: […]
- Justice and Home Affairs (even though the EFTA countries are part of the Schengen area); […]
“Justice and Home Affairs” is an old name for what used to be called the third pillar of the EU, under which the Schengen regulations fall (hence the parenthesis). Schengen or open borders are not mentioned anywhere in the EFTA Convention, the Short Overview or Annexes either.
I could not find a reference for that in a quick search this morning but as I recall, Switzerland was very keen on getting into the Schengen area (in particular to make it easier for tourists touring Europe to include Switzerland in their itinerary), it's not something that was forced on them as a condition to get the rest.
But there is a limit to the weight of such legal considerations in this context. If it becomes expedient, it would be reasonably easy to find a way around such a requirement. Conversely, accepting a new member to the EEA requires an enlargement agreement ratified by all current EU and EEA members (thus giving them some leverage to insist on arbitrary new requirements). Basically, this will be handled at the political level.
On the other hand, whether an EFTA country becomes a full EEA member or tries to achieve the same thing through bilateral agreements (the Swiss model), access to the single market depends on the free movement of persons. To date, there has never been any exception to that because it's extremely difficult to see how it would work without unraveling the whole construction, especially with a country as big as the UK. Freedom of movement (for goods, services, capital and workers) is what the single market is about since the beginning, not a side project or a mere legal requirement.