Will lack of IPv6 be a short-term problem for you: probably not
But the internet is moving towards IPv6. Take a look at what Google sees. When writing this the graph has just peaked to 12.5%. That means that 1/8 of the internet has IPv6 connectivity. Because IPv4 addresses have run out [1] many of those users will be behind some central NAT device. Analysis from Facebook has shown that customers that can load their news feed over IPv6 see a 20% to 40% shorter load time than those that are forced to use IPv4.
So for a certain group of your users making your website accessible over IPv6 will give them a significant speed boost. And that group of users will grow very quickly. Even if you don't want to bother with IPv6 right now, it will become important soon.
Because all of this, when choosing a vendor (whether it is for equipment, services or hosting) it might not be very wise to pick one that doesn't support IPv6. At least make sure they are working on it and can give you a timeline in the contract. Otherwise you might need to migrate to a different one that does properly support IPv6 later.
[1] You can but some on the market, and some regions have a few addresses reserved for newcomers, but for all practical purposes it has run out