My mother just passed away, her home is mortgage free and owned free and clear by her. She left everything to us 3 kids. We would need to split the home value 3 ways. Myself being the oldest would like to purchase the home. How do I obtain a mortgage and be able to give my sister and brother their shares that is left over after getting the home.

Was there a WIll... and did this go through probate yet? What country and state was this?

The easy way to do this is if the value of her estate is at last three times the value of her home (in other words if she has significant investments or other property) you could work with the executor to split her estate equitably three ways with you taking the house and your siblings getting enough other assets to make up for the third they would otherwise be due. But if the size of her estate doesn't allow that, then each of you gets 1/3 of the house, the executor titles it as joint property, and then you work with a mortgage lender to see about buying them out. You should have an attorney draw up the purchase and sale agreement, and I would strongly suggest that you get a professional appraisal of the property so there is no doubt as to what the price ought to be.

You can buy out your siblings before it finishes probate or after. The probate court will want to know which, and you need to be clear on whether you can do it.
Yes, you need to pay for an appraisal (not just an appraisal done by a real estate person) so that there is no legal dispute with siblings.
If you don't think you qualify for even 2/3 the value of the house, your siblings can write you a mortgage (if they are willing). They earn a decent interest and you save a little bit over bank rates, plus you save a LOT on upfront costs.
You should have a lawyer handle it. If they don't want to write a typical 30 year term, they can base it on 30 years but make it payable at the end of far fewer years, giving you time to hopefully gain some equity and value and get a mortgage from a bank.