Can a person go to the VA and try to get disability when they do not have their military medical records?

Damn unlikely, since they will want your records before making a decision.

A1C Cheryl Post You can go through the VA or do what I did use the American Legion National HQ to help you get your med records and they can also help you with your disability claim. God bless you.

Absolutely. While already having your records will speed the application process, the VA has a duty to assist you in getting them.
A couple of things you should note before you start though -
First, your disability rating, if awarded, will take its effective date from the date you file a notice of intent with the VA, so long as you complete and submit your completed application within 1 year. That means that any disability benefit you may be awarded will be back-dated to the day you file the notification of intent -- which could get you up to a whole year of back-pay.
Second, do your homework. If you have a disability you think is due to your service, make sure you can provide evidence to corroborate that belief. If you were treated for it while in service, there should be a note of it in your medical records but getting statements from people you served with who were aware of the problem at the time, as well as doctor's opinions on the cause of your problems will go a long way towards helping you get a positive result. Read the CFR governing VA claims and really get to know your stuff so that when you are examined you can tell them what they need to know about your condition. Look to see if there are DBQ's (disability questionnaires) specifically for your problem and see if your doctor will complete them for you.
Final note - Get ahold of someone at Disabled Veterans of America and ask for help. Don't be suckered by people promising to get you this or that for a portion of your payout. DAV will help you for free and they have credentials from the VA while many fly-by-night lawyerly types do not.