Carla Stefaniak, 36, disappeared on 28 November, the day she was due to leave from San Jose after a birthday trip with her sister-in-law.
In her last message, she told a friend the power was going out at her rental, adding: "It's pretty sketchy here."
Her family are looking for her in Costa Rica, and say she is in "imminent danger".
Ms Stefaniak, from the Miami area, and her sister-in-law April Burton were visiting Costa Rica to celebrate Ms Stefaniak's birthday, but Ms Burton returned to Florida a day before Ms Stefaniak.
The women had stayed at an Airbnb rental in the San Jose suburb of Escazu.
The last time Ms Burton saw Ms Stefaniak was when she dropped her off at the airport on 27 November.
Ms Stefaniak then reportedly took an Uber from the airport to do some sightseeing around San Jose before she was dropped off at the Airbnb.
She had been actively messaging friends and family throughout the day.
Her final text to a friend around 21:00 local time indicated it was raining hard and the power in her Airbnb accommodation had gone out.
Family and friends texted Ms Stefaniak for her birthday on 28 November, but received no response.
Ms Stefaniak had a flight scheduled for the following day, for which she had checked in and scheduled an Uber.
But her family later learned she never boarded the flight home.
The Airbnb rental owner and local security guards said they saw her get into an unknown car with her luggage at around 05:00 local time, according to the family.
"None of us really believe this 5am story because it really doesn't make sense," Ms Burton told Fox News.
"We know she was abducted. There's no reason for her not to contact anyone."
According to the family's Facebook page, Finding Carla, officials "will not pursue the matter" because Ms Stefaniak appeared "rational at the time of her disappearance and got into a taxi of her own free will".
Airbnb has declined to comment on the matter.
The US Embassy in Costa Rica did not immediately return a request for comment.