Lee, who secured a place in the team through her world ranking of 27, made her way on to the stage for Thursday's opening ceremony at the women's version of the Ryder Cup despite not having eaten for almost 48 hours.
Inkster had planned to select the 20-year-old for Friday morning's opening foursomes but has instead had to pair Brittany Lincicome with Michelle Wie in match number two against English pair Charley Hull and Melissa Reid.
"She just can't shake this stomach bug," Inkster told reporters at the St Leon Rot Club. "They gave her an IV (intravenous drip) of fluid today but we'll just see how it goes.
"She thinks she's getting better and as soon as she starts eating something it doesn't sit well with her. I don't want her heave-hoing out on the golf course, it's not fun.
"I don't want to put her out there when something like that is going to happen. We're a man down right now and everyone on the team knows it," added Inkster.
"Everyone is going to have to step it up a little bit, somebody might have to play all five matches now. I didn't want to do that...we're just going to have to play it by ear."
Unlike the Ryder Cup, however, Lee will forfeit a point if her illness continues and she is unable to take part in the final-day singles on Sunday.
In the men's equivalent team event, a half-point goes to each side if a player is forced to miss the singles through injury or illness.
Europe's Solheim Cup captain Carin Koch suggested she herself would soldier on and play through an ailment if she had to.
"I've had a stomach bug before and I've played without eating for a few days," the Swede said.
"If you have to play, you go out and play. All of us are hoping she will be fit to play at some stage before Sunday, for the sake of the matches.
"She's young. If you're young you recover pretty fast."
(Editing by Ed Osmond)
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