Did Bill Ferny and his gang kill any Hobbits when they took over and exploited the Shire?
Only one Hobbit (Lotho) was noted as killed prior to the Battle of Bywater and the uprising:
'Worm killed your Chief, poor little fellow, your nice little Boss. Didn't you, Worm? Stabbed him in his sleep, I believe. Buried him, I hope.'
They just locked up the other Hobbits, probably to avoid a full scale uprising as was already happening by the Tooks. All those who were noted as being arrested came out of Michel Delving worse for wear but alive.
Old Will the Mayor set off for Bag End to protest, but he never got there. Ruffians laid hands on him and took and locked him up in a hole in Michel Delving, and there he is now. And after that, it would be soon after New Year, there wasn’t no more Mayor, and Pimple called himself Chief Shirriff, or just Chief, and did as he liked; and if anyone got “uppish” as they called it, they followed Will.
During the Battle of Bywater, 20-ish Hobbits were killed, a couple during initial skirmishes:
‘Whips, knives, and clubs, enough for their dirty work: that’s all they’ve showed so far,’ said Cotton. ‘But I dare say they’ve got other gear, if it comes to fighting. Some have bows, anyway. They’ve shot one or two of our folk.’
‘There you are, Frodo!’ said Merry. ‘I knew we should have to fight. Well, they started the killing.’
(Merry's comment also lends credence to these being the first deaths.)
Another nineteen were slain in the Battle itself:
At last all was over. Nearly seventy of the ruffians lay dead on the field, and a dozen were prisoners. Nineteen hobbits were killed, and some thirty were wounded. The dead ruffians were laden on waggons and hauled off to an old sand-pit nearby and there buried: in the Battle Pit, as it was afterwards called. The fallen hobbits were laid together in a grave on the hill-side, where later a great stone was set up with a garden about it. So ended the Battle of Bywater.
And, as Daniel mentioned, two more Hobbits were killed in Bree:
there was trouble right here in Bree, bad trouble. Why, we had a real set-to, and there were some folk killed, killed dead! If you'll believe me.'
'I will indeed,' said Gandalf. 'How many?'
'Three and two,' said Butterbur, referring to the big folk and the little. 'There was poor Mat Heathertoes, and Rowlie Appledore, and little Tom Pickthorn from over the Hill; and Willie Banks from up-away, and one of the Underhills from Staddle: all good fellows, and they're missed.'
If you include Wormtongue as a henchman of Saruman, then technically the answer is yes. Saruman implies that he murdered Lotho Sackville-Baggins
Wormtongue halted and looked back at him, half prepared to stay. Saruman turned. 'No evil?' he cackled. 'Oh no! Even when he sneaks out at night it is only to look at the stars. But did I hear someone ask where poor Lotho is hiding? You know, don't you, Worm? Will you tell them?'
Wormtongue cowered down and whimpered: 'No, No!'
'Then I will', said Saruman. 'Worm killed your chief, poor little fellow, your nice little Boss. Didn't you, Worm? Stabbed him in his sleep, I believe.
It is also suggested that Saruman used the power of his voice to get Wormtongue to actually do this, but technically he was murdered by the hand of Wormtongue so I would say the answer to your question is yes.
In addition to Lotho, two hobbits were killed during the Battle of By water, described in The Scouring of the Shire.
Also, although it doesn't count as the Shire, two more hobbits - and three men - were killed earlier in Bree, according to Butterbur.