I'm looking to get into Dungeons & Dragons 5e and playing with other players online through roll20.net and other sites, but I wonder if Player's Basic Rules will be sufficient for me in order not to disturb the game in any way until I get the Player's Handbook. At the time I have no interest in GM-ing, so that won't be a problem for me, by the way. Thank you!
The basic rules are comprehensive for basic gameplay, but lacks a ton of content that would improve your gameplay experience. If you had access to Player's Handbook for building your first level character and have copied the relevant bits on spells and character options for reference, you can play with the basic rules just fine until the second level, at least. Otherwise, you'll be fairly limited from the very start.
Many character options are absent and this may become a problem as your character levels up or if you have to build a character using the basic rules only. The basic rules contain only four classes: Cleric, Fighter, Rogue and Wizard. Each class only has one archetype: Life, Champion, Thief and Evocation, respectively. These are not particularly weak choices, but you will lack freedom of choice in deciding how to specialize your character. Rare races are also absent: the basic rules only include humans, elves, dwarves and halflings.
The spell list has also been cut significantly from the Handbook version. There is still plenty of spells available, but not as much variety as with the Handbook.
The optional rules on multiclassing and feats are missing, too, but multiclassing is unlikely to be a concern to you until you get the Handbook, and feats only matter for the Variant Human race pick.
Other alternatives for the basic rules
The System Reference Document (SRD) has more content than the basic rules, released under the Open Gaming License (OGL) for players and third-party content developers. You can read more about it on WotC's website.
Other possible extensions for your character options, available without the Player's Handbook, include the Unearthed Arcana playtest materials. However, they are officially not yet finished so they may be imbalanced or play poorly - always ask your GM before bringing such characters into play. If your GM is inexperienced, I would recommend against using them.