Yes. My family has purchased a number of instruments online: Banjos, acoustic guitars, solid body electrics, drum sets, a bass or two, and they've all arrived in pristine condition. There are plenty of websites that sell quality musical instruments. Amazon is an obvious one, but there's also
Music123, and
Musician's Friend.
Since you're pursuing the ukelele, you can check out this page:
Music 123 Ukelele Selection. Since you're looking for budget, and you're a beginner, I recommend any of the Rogue series of Ukeleles. Here is a nice one and with a nice price to go with it:
Rogue Hawaiian Soprano Ukelele. Rogue is a solid starter brand, that is flexible enough to extend into intermediate practice. Unless you're a top of the line pro, you shouldn' pay more than $50 for a decent Ukelele. Plus, right now, Music 123 is offering free shipping on much of their inventory (we always opt for the free shipping when we can). I say this because you can go ultracheap, but you may end up with a poor sounding paperweight, as bottom-end instruments can lead to frequent detuning, poor sound, and shoddy construction. Something like that Rogue, though, would be about right.