Of course- but what kind? You definitely want a DSLR (Digital single-lens reflex camera). As long as it isn't a tiny digital rectangle camera and has a detachable lens, you can pretty much assume it is a DSLR. I have always had Canon cameras, but Nikon, Sony, and Fujufilm are some other big names in the camera world. My first camera was the Canon Rebel Xti with the kit lens (I'll explain this more later). It was a decent camera, but not one that I would recommend to anyone purchasing one new as it used a CF card instead of an SD card which are rarely made anymore. If that is all you can afford, it will still do a good job for you, but if you want to expand into more advanced photography, it will have its limitations.
So, firstly look for a camera that takes an SD card. I currently use a Canon Eos Rebel SL1 which I bought used from a photographer friend who was upgrading. If you can afford this nicer body it will prevent you from wanting to upgrade in the near future. This one has a touch screen viewing panel and allows you to see the images in the viewing panel rather than always having to look through the view finder when photographing.
I originally bought a small canon camera bag that I quickly grew out of. Now I have this gorgeous bag from Johansen Camera Bags (Jo Totes). I have their Georgia Nautical bag and it is just perfect. The inside has velcro sides and it comes with dividers so you can decide how large you want the interior divisions. I can hold everything expect for my larger tripods and still have room for my keys, wallet, sunglasses, snacks, etc.
Now your personal style will affect your camera bag choice. You may want something small and compact to house your camera that you can throw into a diaper bag. Either way, you want something specifically designated for holding your equipment to keep it safe.
After 2 years, I am just now realizing how important your camera's lens it! I have always taken my photos with my kit lens. The kit lens refers to the lens that comes with your camera and is usually very limited. The kit lens for most Canons is an 18-55mm f/3.5-5.6. I'm not going to go into a lot of detail on what that means, but the gist is that this lens does fine when you are outside and have plenty of natural light, but struggles indoors or in the evenings. When you don't have a ton of light, your images will turn out noisy (grainy) and any movement will cause blurs (not ideal for kids!).
Recently I upgraded to the 50mm f/1.8 and wish I had done this from the start. If you buy one open boxed or used, it is only around $100 and will save you so much pain and annoyance down the road. The lens is so much better for shooting indoors and moving objects. If you are tight on budget, the kit lens will do the job, but down the road you should look at upgrading.
Below are some examples of images taken with the kit lens vs. the 50mm so you can see the different in the image quality.
You will also want an SD card. Some cameras might come with one but it will probably be a small GB (amount of storage room available). Think of how annoying it is to delete photos when you phone runs out of space- it is 10x as annoying when this happens to your SD card the second your kid starts cooperating! I would recommend a 16 or 32 GB card. If you are also interested in quickly get your photos from your camera to your phone, you will want to invest in a wireless card. I have theFlashair 32 GB card and love love it. The Flashair connects your camera to your phone and allows you to download the images to your phone without removing the SD card from the camera.
A Lens Strap
You also want a lens strap to prevent you from accidentally dropping your camera while using it. My camera came with a canon strap but I wanted something cuter! My strap is from Aspen Willow's Etsy shop. I don't think she has this exact one right now, but they are so soft and light. Buying a cute one isn't necessary, but you do need to have some kind of strap.
That last thing you need is something to edit with. That can be a free app like VSCOcam or a purchased program like Lightroom. I currently use a mixture of VSCO filters HB1 and HB2 and Lightroom (you can purchase Lightroom outright or pay a monthly subscription of $9.99). In Lightroom I use Pretty Presets and have thepretty film collection.
There are a few things that you may want to invest in but aren't necessities. A tripod is nice if you want to be in photos with your children. If you don't plan on every being in front of the lens, then skip this. If you want in the photos, a remote clicker is also nice because you don't have to set your camera to a timer and run back and forth. I purchased a bundles of 3 tripods and a remote clicker from Amazon that I would definitely recommend.
And that is it! In summary, you need a camera, a lens, a strap, a bag, an SD card, and something to edit with. If your camera comes with a lens, a strap and an SD card, you don't have must else to purchase. I have currently invested a total of $450 dollars in my equipment and editing programs (I sold my original camera so my upgrade didn't cost much out of pocket) which is totally doable if you save up.
Time to ask for your Christmas present!!