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Early voting is underway in Austin on Proposition 1, where residents will decide which regulations the city should adopt for vehicle-for-hire companies like Uber and Lyft.

Both companies have pledged to leave the city if the proposed ordinance is not adopted — a claim they've made good on in three Texas cities this year. But at least one ride-hailing company insists it can fill the gap Uber and Lyft would leave behind.

"We're not going to be the donkey or the elephant," said Jonathan Laramy, the chief experience officer for Get Me LLC, which the company has stylized as getme. "We're here to stay. Vote Prop. 1, vote Prop. 2 – we don't care."

The proposed ordinance in Austin magnifies a conflict that has taken root in cities across the country: the debate over fingerprint background checks for vehicle-for-hire drivers. Officials with both Uber and Lyft have taken issue with these sorts of checks, calling them burdensome and unnecessary. Austin's proposed ordinance, largely backed by Uber and Lyft, would prevent the city from requiring such checks in the city.

Read MoreAhead of Austin Election Over Uber, Local Leaders Take Sides

Laramy said getme — which currently operates in Austin, Dallas, Houston and Las Vegas — is willing to adhere to any local regulations, as long as the process for obtaining fingerprint-based background checks is "fast, easy and cost effective."

"We're a good corporate citizen," Laramy said, adding that the company is willing to collaborate with cities on their regulations.

While his company is still working out the specifics, Laramy said that "at some point, we will fingerprint all of our drivers" — even in cities without a requirement.

If Austin voters do not approve the proposed ordinance, Uber and Lyft have said they will leave the city — although The Daily Dot reported last week that Uber fully intends to stay, regardless of the outcome of the election. If the companies leave, Laramy said getme would be prepared to process a potential influx of driver applications.

"We have a platform where we could actually — and we already have this in place and ready to go — sign up conceivably 5,000 drivers in a month, if not more," Laramy said. He would not elaborate on specifics of the plan, but he said it involved "using information that's already been done and then verifying and showing us that."

After starting up in Dallas in February 2015, getme recently relocated its headquarters to Austin. Laramy said it has more than 10,000 drivers across the four cities where it operates, more than 2,000 of whom are in Houston. The company boasts 6 corporate employees and a handful of contractors, making it a significantly smaller operation than ride-hailing giants Uber and Lyft.

Laramy says the company soon plans to offer services in Los Angeles, Denver, Chicago and Atlanta. In Texas, he said, the company is launching operations in Galveston next week and Corpus Christi this summer.

This follows Uber's cessation of operations in Galveston and Corpus Christi earlier this year after both cities adopted fingerprint background check requirements. Laramy said getme's interest in both cities was unrelated to Uber's actions and that they had planned to launch in both locations well before Uber left.

"You can't get home if you take a ride down there," said Laramy, describing someone looking to travel between Houston and Galveston using getme. "It's silly not to have both cities."

Read MoreAcross Texas, Uber Puts Cities in Tough Spot

The company announced Monday that its application for a chauffeur's license in Galveston was approved — opening the door to do business on the island. Galveston spokeswoman Kala McCain said the city regulates all transportation companies the same way.

"We had existing ordinances in place for ride-sharing companies that applies to not only getme and Uber-type services but also our taxi cabs as well as our horse carriages and our pedicabs," she said. "They all fall under the same ordinance because they're all providing public transportation for hire in some form or fashion."

Galveston's ordinance also requires drivers to undergo fingerprint background checks, which Uber said prompted it to leave. McCain said the city did not have specific numbers on how many drivers Uber had working in the city, as many of them were operating without the required chauffeur's license.

"We did get quite a bit of pushback on social media," she said. "People being really upset about the fact that we did not allow Uber to operate, which is obviously not the case. They didn't want to adhere to the city ordinance that was in place."

Disclosure: Uber and Lyft have been financial supporters of The Texas Tribune. A complete list of Tribune donors and sponsors can be viewed here.

Correction: An earlier version of this story misspelled the name of Galveston spokeswoman Kala McCain.