Lets break down things you can do to minimize the friction, and hopefully sway the DBAs in helping you out in a meaningful and productive manner.
The Request itself
How you request things is arguably the most important. You should request information as soon as you know you need it (If it's something where you'll need it later, because you'll need up to date information, you should simply request that you will need whatever data at XX time) The sooner you can request this the better. (If they are prone to forget send them reminder the morning / a bit before you need the data)
You need to use their process. If they don't have a set process ask them how they would like you to make these requests. Ultimately if you aren't provided with a means of getting the data yourself then they (or someone on their behalf) needs to be responsive to fulfilling these requests.
Be specific! If you ask for "all sales data" the DBA is NOT going to be happy... If you ask for The customer's name, amount paid, item[s] purchased, and timestamp of the transaction from 10/1/2014 to now they will be far more amenable to the request. (basically make sure they know exactly what you need, vague requests can be VERY frustrating)
DBAs and Priority
DBAs work a position where there is ALWAYS more to do, and never enough time to do it. As such the typical DBA attacks problems purely by priority. What dictates that priority varies person to person, but typically it's either whatever has the most significant impact to the company (Increased income, Mitigate risk) or Whatever the person over them says is priority.
Sadly it's a near certainty that data request with no real context to it's value is going to be low priority. In the ever growing list of things that need doing the low priority list tends to be a grave yard where requests go to die.
So depending on what the priority system is (you can ask, most will just tell it like it is) you need to tackle it to make it clear the value of your request.
If the DBA just follows the chain of command you need to get their boss sold on the fact this data is important. Perhaps it's for a sales campaign that could potentially be highly lucrative, or perhaps your CEO said you need to get them to do this. Once you can say "Hey DBAs I need (Specific Request) CFO Chad wants to get a campaign going that could be good for business, when can you assist?"
You just showed the DBAs this has a monetary value to the company, and the execs want it. You just made your request more valuable and thus a higher priority! (Don't just always name drop an exec, you're not trying to drum up your requests to priority one, you're trying to show that these requests have an appropriate value and you're not just asking for the sake of asking)
You're one of us, not them
IT at large tends to be a tightly knit group. They tend to hear all the complaints with little praise and work closely "Fixing your problems". Often this results in a Us or Them mentality where you're either part of their team, or you're those annoying end users. (This is a generalization, many IT dept have a much healthier outlook than this)
You want them to feel you're as much their team as anyone else. If you need to throw a short deadline for a request to the DBAs don't say "Hey DBAs, I need X ASAP" That's something "one of them" would do. Instead say "Hey DBAs, I'm really sorry for the short notice, but the CEO just asked me to X and Y, but I need data Z to do it. Any chance you could help me out here?" This time you aren't one of "them" you're as much a victim as they are, they'll be far more amenable to trying to bail you out. (as long as this doesn't happen all the time)
Build a friendly relationship with IT. Most people just don't network around the office, Seriously networking with your support staff is a HUGE benefit. Simply put when things go to hell IT can save you from some of the most crushing mistakes, or they can let you drown in those mistakes. (not that they do so intentionally, but people are more likely to bail you out if you have a healthy working relationship)
Bribes are one of the single most effective tools to getting IT on your side. There is also no better work appropriate bribes than food! IT doesn't work in a "What's in it for me?" manner, but again if you have a good working relationship it's far more likely I'll bump you up the priority list. (Is this how it should work... probably not, but it also has it's perks)
In the Us or Them situation, people within IT might pick each other up a lunch, or say a heart felt thank you, etc. Because seriously IT is a really thankless jobs where the vast majority of users frankly dump all over IT because their preventing large scale problems are often inconvenient to individuals.
Don't be another thankless user dumping working on IT then responding with an email that just says "thanks" that's seriously the least you could do for someone who took time out of their day to help you out (even if it's their job)
When you pass them in the hall stop and say "Hey Bob, Seriously thank you for Wednesday, my boss was really riding my butt, you're a life saver" or if your company does some kind of kudos or shout out (Basically where one employee / department recognizes another) make it a point to put one out there from time to time. If IT is mostly secluded to their "cave" make it a point to drop by and thank them in person.
It IS their job to help the users, but seriously most IT depts listen to nothing but complaints, whining, and demands all day. Getting that occasional REAL thank you. (and if you're really awesome something to snack on) Really helps take the edge off their day, and if you come by and make their day a little easier, they will almost certainly return the favor.