When the Yellow Vest movement exploded in November, its grass roots character was key to its success, giving voice to people who felt the establishment—from unions to political parties—were out of touch with working-class pain. But now, as it held its 15th consecutive weekly demonstration on Saturday, the movement is divided about how to leverage its popularity before it loses momentum in pressing President Emmanuel Macron for change. Ingrid Levavasseur, whose passion and moving personal story made her something of a celebrity at protests, exemplifies the conflict over whether to coalesce into a formal organization or keep its antiestablishment roots. Ms. Levavasseur, 31, was participating in a Yellow Vest protest this month when fellow demonstrators turned on her, pulling her hair and insulting her. Her offense: Her decision to run in European Parliament elections in May. "Take off your yellow vest," shouted the crowd. Riot police had to escort her out of the crowd. Ms. Levavasseur, a soft-spoken single mother, emerged as a prominent figure very soon after the protests began. One of four siblings, she grew up with an alcoholic father. The family was often homeless, living in shelters and eating in soup kitchens. She quit school early to work as a waitress. After a divorce, she passed an exam to become an assistant nurse. "I don't drink, I don't smoke, and I don't have any hobbies. Every penny I earn, I spend it to pay the bills and feed my kids. But it's still not enough," said Ms. Levavasseur, who raises her two children, 8 and 13, on her own. She had never attended a protest before joining the first demonstrations in mid-November near Val-de-Reuil, her hometown in Normandy. More than 250,000 people took to the streets across France. "It was a revelation," said Ms. Levavasseur. "Suddenly, I wasn't alone." Later, after violent clashes broke out in Paris, the French government tried to negotiate with the yellow vests, but struggled to find any leaders. Two members responded to an invitation by Prime Minister Edouard Philippe to meet. The first came, but refused in the end to meet Philippe, saying he'd received death threats from other yellow vests opposed to the meeting, said a senior government official. The second left minutes after he arrived, refusing to take part unless it was filmed. Since then, some members of the movement have criticized would-be representatives for the group as pursuing their own personal agenda. Others began thinking about the next step for the movement, holding meetings across the country. In early January, about 200 yellow vests met in Marseille to discuss the possibility of organizing a political party. Bernard Tapie, a business tycoon and former politician, provided the venue and paid for the security. Outside, more than 500 yellow vests protested. At that meeting, Ms. Levavasseur met several other yellow vests with whom she would later form a new group to run together in May's European elections, a race that is likely to see a surge in antiestablishment parties across Europe. But a day later, when news got out that Ms. Levavasseur had been offered a lucrative job as a TV columnist at a leading national TV channel, she received an avalanche of insults and threats on Facebook. "I got scared," said Ms. Levavasseur. She turned down the offer. — Agencies