And of course personally, I don't think Legolas had any siblings of any sort at all. I just don't think it's a simple 'no, not possible' type answer. Just trying to get us all to expand our minds more, and something that isn't ever told or said, doesn't necessarily mean it's not a possibility.
Or, Thranduil could figure that he's going to sit back in his cave for the rest of his life and just say in Mirkwood, therefor there's no point in having an heir...(which he seems to do at this point in LOTR)?
But, Legolas was rather young for an Elf, and this wouldn't seem something of dire need especially with Dol Guldur active and breathing down Thranduil's neck. Now, Boromir, heir and eldest of Denethor's was sent to Rivendell, but he forced his way until daddy said 'ok,' was getting answers to a puzzling riddle, and also there was always Faramir.