They committed crimes other than using Unforgivable Curses.
Over the course of their attempt to take over the wizarding world, the Dark Lord and the Death Eaters did commit other crimes known to carry sentences in Azkaban that were unrelated to the Unforgivable Curses or the Dark Arts. Some of the Death Eaters were actually convicted of things other than using Unforgivable Curses.
Breaking into the Ministry and attempted theft:
Sturgis Podmore was convicted and sentenced to six months in Azkaban for trespass and attempted robbery after breaking into the Ministry.
“Sturgis Podmore, 38, of number two, Laburnum Gardens, Clapham, has appeared in front of the Wizengamot charged with trespass and attempted robbery at the Ministry of Magic on 31st August. Podmore was arrested by Ministry of Magic watchwizard Eric Munch, who found him attempting to force his way through a top-security door at one o’clock in the morning. Podmore, who refused to speak in his own defence, was convicted on both charges and sentenced to six months in Azkaban.”
- Harry Potter and the Order of the Phoenix, Chapter 14 (Percy and Padfoot)
Some of the Death Eaters broke into the Ministry and attempted to steal the prophecy - a very similar situation to what Podmore was convicted for.
Setting dangerous creatures on people:
Hagrid was thrown in Azkaban for allegedly setting a creature on students. No one died from it, but setting the creature on students was enough to be considered a crime and worth a sentence in Azkaban.
“They think, up at the school, that Hagrid’s been setting a – a – something on students. They’ve taken him to Azkaban.”
- Harry Potter and the Chamber of Secrets, Chapter 15 (Aragog)
The Dark Lord had set Nagini on Arthur Weasley, and as a student at Hogwarts, the basilisk on students, resulting in the death of Moaning Myrtle.
Attacking Muggles:
Attacking Muggles seems to be a crime that carries a sentence in Azkaban, however it’s done. Morfin Gaunt was convicted for making Tom Riddle break out in hives - clearly not one of the Unforgivable Curses.
“Ogden Apparated back to the Ministry and returned with reinforcements within fifteen minutes. Morfin and his father attempted to fight, but both were overpowered, removed from the cottage and subsequently convicted by the Wizengamot. Morfin, who already had a record of Muggle attacks, was sentenced to three years in Azkaban. Marvolo, who had injured several Ministry employees in addition to Ogden, received six months.”
- Harry Potter and the Half-Blood Prince, Chapter 10 (The House of Gaunt)
Attacking Muggles seems to get two different charges - violating the Statute of Secrecy by doing magic in front of Muggles, as well as causing harm to the Muggles.
“He glanced down at his scroll of parchment again. ‘Morfin will attend a hearing on the fourteenth of September to answer the charges of using magic in front of a Muggle and causing harm and distress to that same Mugg—’”
- Harry Potter and the Half-Blood Prince, Chapter 10 (The House of Gaunt)*
A group of Death Eaters attacked some Muggles during the Quidditch World Cup without using any Dark magic - they levitated them.
Leaking Ministry of Magic secrets:
One Death Eater, Augustus Rookwood, was convicted of telling the Dark Lord the Ministry’s secrets.
“Augustus Rookwood, said the caption beneath a pockmarked man with greasy hair who was leaning against the edge of his picture, looking bored, convicted of leaking Ministry of Magic secrets to He Who Must Not Be Named.”
- Harry Potter and the Order of the Phoenix, Chapter 25 (The Beetle at Bay)
This didn’t involve any Unforgivable Curses or violent crimes, but it got Rookwood sentenced to Azkaban for what seemed to be several years. This would likely still be a crime even if neither the Dark Lord nor the Death Eaters used Unforgivable Curses - it’s probably a crime to leak Ministry information to anyone who’s not supposed to know it.
Being an unregistered Animagus (Wormtail only):
Despite three of the Marauders doing it, it’s actually a crime to be an unregistered Animagus.
“Otherwise, as you very well know, I will inform the authorities that you are an unregistered Animagus. Of course, the Prophet might give you rather a lot for an insider’s account of life in Azkaban.”
- Harry Potter and the Order of the Phoenix, Chapter 25 (The Beetle at Bay)
Only Wormtail was an unregistered Animagus, but he could be sent to Azkaban for it.
They perhaps could legally rule the Ministry by changing strategy.
The Dark Lord could possibly legally rule the Ministry - the Ministry only has to answer to the International Confederation of Wizards, and other than that, the Minister seems basically able to do what he wants apart from breaking the Statute of Secrecy. However, he’d need other Ministry wizards who were willing to go along with him, otherwise he’d just be removed as Minister. It’d likely be more difficult to get enough of his supporters legally among the ranks of the Ministry - to legally take it over he can’t use the Imperius Curse, so the Death Eaters would need to actually become Ministry employees. The Dark Lord would need to legally get enough of his supporters into the different departments before he’d be able to start making drastic policy changes, so he could remain in power.