Vtech's "official" position is that the band is not a replaceable part. They don't sell them on their website.
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=== Update (04/03/2017) ===
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Final confirmation. You CANNOT purchase new wristbands for the Kidizoom Smartwatch.
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From VTech:
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"Based on your description, you have the Kidizoom Smartwatch. Again, please be advised that the wristband is not a replaceable part, thus it is not available on our website to purchase. Note that the warranty for your toy is 3 months from the date of purchase and as such, yours has passed the warranty timeframe. You may want to take advantage of promos and discounts offered by our partner retailers. You can visit www.vtechkids.com to learn more about our newest products. You will also be redirected to our partner retailers where you can purchase VTech toys."