Just peeked at FamilySearch. They have a huge catalog — separate from the simple record search on the site — and they are the best bet for finding worldwide records without contacting or going to the country directly.
They seem to have some 20th century records from some Ecuadorian places microfilmed. But, what I see is only available in Utah at the huge library there. If a trip to Utah is easier for your friend than a trip to Ecuador, she might reach out to the librarians there to see if there is a reasonable pathway for her to get some useful information through a visit.
Otherwise, if she knows the province in Ecuador, she could reach out directly to the archives (I recommend writing in Spanish) to see if they could do a lookup or help her with a visit.