You probably don't want to hear this, but it is really normal that your baby isn't sleeping through the night - think of all the things her mind and body is processing. And as far as her grandparents taking naps with her, it is awesome that you have parents who are willing to love and cuddle your daughter instead of sticking her in a crib somewhere so they can get things done. You and your daughter are truly blessed to have that. The picture of grandparent and baby napping together is precious!
This whole thing about "teaching" babies how to sleep is confusing to me and personally I don't subscribe to the theory behind it. How can it be that babies need to be taught to sleep - we are humans, we know how to sleep, we just sometimes have needs that happen during the night (have you ever gotten up to go pee in the middle of the night or been awake for no reason?) What if I told you, "Now you are going to sleep." Unless you are tired and in the perfect mood for sleep (which it sounds like you are!), you are not going to just fall asleep. It doesn't work that way. If it did, there would be no such thing as insomnia. Sleep is not an on or off thing like a switch. Babies know what they NEED and for some, they need more comfort and closeness, not to be "taught" to not bother mommy or daddy while it's dark out. Parenting is not just a day time job, but a night time one too. And, babies have so many things going on with all the skills they are learning - mentally and physically - and just regular growth that sometimes they're going to need more of your attention (daytime and/or nighttime) - teething, growth spurts, milestones (which your baby is right in the age for heightened mobility milestones), etc. By leaving your baby to cry, you are not really teaching them to sleep on their own, but what you are teaching them is that you are not available to them 24/7 and that if they are upset (crying until vomiting), you cannot be bothered. That is why babies sometimes "learn" to sleep with this method. If no one comes to get them when they need it, they learn to not expect that love, care and concern. And yes, having your baby think this would give you a nice break at nighttime, but think about what it does for your relationship and the trust/love that you share and for their self-esteem.
I know it's not easy, but hang in there. This time is so fleeting and when it's over you will be so happy and proud of yourself that you stayed with it and gave your daughter EVERYTHING that you could. Isn't that our job as parents? It truly takes a strong mama to just be there for her baby without trying to "train" it to be less inconvenient. Good luck and best wishes :)