The trade body also reported "worrying" data suggesting the UK manufacturing sector could go into recession.
Eleanor Temple, chair of R3 in Yorkshire and a barrister at Kings Chambers in Leeds, said: “There’s been some worrying national data in the last few weeks indicating that Brexit uncertainty may be pushing the UK’s manufacturing sector towards recession.
“After a resilient performance in the run-up to the original March deadline for leaving the EU, it appears that the stockpiling which masked falling confidence is now over, with many businesses reporting weak investment and shrinking order books."
She said this, together with concerns about the risk of increasing barriers to trade around the world, provides a "fairly gloomy" outlook.
"It appears that manufacturers in Yorkshire are faring relatively well. Indeed, the region as a whole remains fairly buoyant with over half of the sectors surveyed by R3 showing month-on-month decreases in elevated risk," she said.
"However, with the UK continuing to face a lack of clarity about the nature of the Brexit deal, there are still some tough times ahead for businesses – so make sure you seek professional advice at the first signs of financial trouble.”
Of the 11 sectors surveyed by R3, seven saw a monthly fall or no change in the level of companies in the elevated risk category in Yorkshire.
Tourism operators experienced the largest fall in companies judged to be at greater than usual risk between May and June at minus 1.6 per cent.
The agriculture and restaurant sectors had the largest rise, although this was an increase of just 0.5 per cent.
The manufacturing industry in Yorkshire experienced relative stability in terms of the proportion of companies in the sector at above-average risk, with the percentage at 38.2 per cent unchanged from last month.
R3 uses research compiled from Bureau van Dijk’s "Fame" database of company information to track the number of businesses in key regional sectors that have a heightened risk of entering insolvency in the next year.