This photo looks vibrant because the photographer made it vibrant, either through a filter (smartphone) or via editing. The camera did nothing but take the photo, the same as your good camera and lens can.
The difference is the photographer. This image has good light, an obviously well planned location and props, a photographer who is engaging the child, and finally, aggressive, but somewhat pleasing post-processing. The white balance looks toward the yellow/green end, but believable, and the exposure of the scene looks good. This is a reasonably well composed shot, but the cropping bothers me as too close on the right side. The child is centered and eyes align with rule of thirds.
Unfortunately, the most impactful thing with the image wears on you after a while: the really aggressive boosted vibrance (basically saturation without impacting skin tones) and perhaps overdone whites really grab your attention, like the over boosted TVs in the TV store.
Of course you can !! In fact if you have a 50mm prime lens, you can capture even better picture than this. This picture is not that much sharp as much a 50mm prime lens can give. Just keep in mind to set the depth of field as you required and you always need some post-processing for making colors in the picture look vibrant ( those are simple lightroom steps ).