Rathlin Energy wants to expand their West Newton A site in Holderness, with the new wells drilled up to 2,000m deep to hit Permian reservoirs which may hold petroleum, a combination of oil and gas.
They also want to “test, appraise and produce” from the two existing wells at the WNA wellsite near Sproatley.
Documents submitted by Rathlin state they will use acid in the process, but insist it is not an “acid fracture stimulation”.
However campaigners believe they will use unconventional techniques.
Fossil Free Yorkshire are distributing thousands of leaflets and are urging residents to make their feelings known, saying the more people that submit objections, the more chance it will be stopped.
Helen Wright, from New Ellerby, said it would be “total hypocrisy” to declare a climate emergency - as East Riding Council did in February - then support the drilling of many new oil wells producing fossil fuels for up to 25 years.
Yorkshire Party councillor Andy Walker said planning to extract more fossil fuels was “insane”, adding: “They obviously can’t see what’s happening on the west coast of America or last year in Australia.”
Rathlin argues that there will be a need for oil and gas as part of the transition to the low carbon economy.
About the process it will use on the existing wells, it says: "The primary approach is to utilise acid to treat the wells below the formation fracturing pressure.
"For the avoidance of misinterpretation, this is not an acid fracture stimulation. This is the same technique used during the exploration phase.
"This technique will be subject to Environment Agency, Health and Safety Executive and Oil and Gas Authority consent."
It says they expect the two wells to make a "seamless transition" continuing to flow from the testing phase into production.
The firm has cut the number of HGVs from a maximum 25 a day to 10. They plan to store up to 490 tonnes of crude oil on the site.
Ward councillor Brian Skow said around 18 months ago he had requested Health and Safety officials from Leeds to visit the site, and they’d found “all in order”.
He added: “Some people think there’s a hidden agenda, I don't think there is, as far as I’m concerned they are carrying on in a correct way.
"They are not a blot on the landscape. They are not like a wind turbine. There is nothing to be seen from the side of the road."