I was recently married and want to take my new last name. I plan to apply for citizenship in the coming months (Been in the US for 15 years), but don't want to have to pay the $500 for an updated Green Card then pay $800 to apply for citizenship. If I change my name with the SS office and DMV, but not my green card will that hinder my ability to be granted citizenship?
If you are applying for citizenship soon (and the fee currently is $680, not $800 - just fyi), you will not need to do a name change because you have already had the legal event occur that permits a legal name change (the marriage), so on the N-400 Application for Naturalization, you simply fill in the first part asking for your name with your married name (not maiden) and provide a copy of your marriage certificate with the application to support the married name. If you input it as a name change, this can delay your swearing-in ceremony because all name changes must be sworn in before a judge to have legal effect (usually the officers at interview will catch this error if they notice it is a married name and correct it for you, but sometimes they do not), so you would have to wait for a court swearing-in ceremony whereas with no name change (as your are already married, this is just a name correction) you just get the soonest swearing-in ceremony they can get you into.
if you do not plan to apply fairly soon, you should probably go ahead and update your green card.
If you change your name with SSA and the DMV, it will not hinder citizenship but could impact your ability to work as your name in the federal verification systems will not match the social security system and then you will have to correct the green card anyway to fix that mess. Until you are a citizen, your green card determines everything else and not the other way around (so you always start with your green card/proof of your current status in terms of correcting information, not the other way around).