You already have a (successful) sleep training answer above, so I’ll give some alternatives to try. My child was a "happy chucker" (ie he vomited a LOT, but didn’t seem distressed by it). About 3-5 minutes of crying would cause vomit, so sleep training wasn’t really a good option for us.
I ended up cosleeping a lot, just me and baby in a double bed set up to be as safe as possible. So baby has their own space in the bed and the same amount of bedding as in a cot. I used one pillow (none on the bed would be safer). I also planned for baby rolling out of bed, but it didn’t happen. He was pretty much in the middle of the bed with me around him, pillows on the floor. Obviously the cot is the safest place but SIDS rates drop off at 6 months and sleep deprivation made me consider the risks acceptable.
You don’t say if your child is in their own room. Sometimes the sound of someone else’s breath is enough, so if they stir and you’re present, they go back to sleep. Sometimes the smell of a parent can help. We had some success with putting my top from the day near the cot.
Lighting is stupidly important. Get a blackout blind so that when you say it’s night, Mr Sun doesn’t contradict you. Where we live, summer evenings are long, so black out blinds are a must.
If you do get up at night, keep it quiet, keep it boring, keep the light levels as low as possible. I didn’t rock or sing, it was cuddles only, or music from the mobile, or milk. I didn’t night wean, but I’ve heard people have some success with that. I did limit eye contact, sometimes, too, but that’s only really necessary when they’re trying to smile and play with you. Even when a full bed change was required, baby would be wrapped in a blanket and ignored while the change was completed rapidly and then back to darkness and quiet and sleep.
I got night nappies (12 hours of dryness!) and didn’t change them during sleep time unless there was poo or a leak. Or they felt really, really wet.
We used appropriate bedding for cosy feet. We used the baby sleeping bags quite a bit, too.
The biggest success I actually had was simply adopting a relaxed attitude to my sleep quota. I stopped looking at the clock. Stopped keeping a tally of how many times I’d been up. Improved my ability to go to sleep quickly (relax face, relax jaw, relax hands, breathe). I was simply as tired as I felt on any given day. We bought one of the color changing clocks, so I could tell "daytime" from "night" without looking at the numbers. And that worked for baby, too, once he was old enough. Stress stops you from sleeping and stressing about how tired you’re going to be is really counterproductive when trying to get to sleep!
My partner helped when required, but I breast fed and I was on maternity leave, so I did most on the night shift (voluntarily).
This will pass, and you might not even notice its passing! Ok, maybe you will, the first few times they sleep all night in their own room, you might lie there, wide awake, worrying about them but too scared to check on them in case you wake them!