If the tax raises all of their costs and it's a competitive market then they'll all have to raise prices, because competitive market = competitive pricing and it has to come from somewhere.
The most common case where it doesn't raise prices is when they don't have any competition and are already charging the monopoly price, so the money has to come from the company because if there was any more to extract from the customer, the monopolist would have done it to begin with. But we don't really like those markets or want to promote their continued existence.
What this doesn't depend on much is the form of the tax. If it's a competitive market then VAT and corporate income tax are both getting passed on to the customer. If it's a monopolist then either one will typically come out of the monopoly rent, because charging the customers more would bankrupt them or cause them to extend the life of used goods instead of buying new ones etc.
However, corporate income tax is generally easier for multinational corporations to avoid than VAT, so they have a pecuniary interest in making people think that VAT is worse than income tax.
There's a third option: they're competitors are small business that don't have the means to do the clever tax optimization/evasions, so they were already paying full tax. So now Starbucks if it were to pay full taxes, would have to either raise their prices (but then competitors could stay at the same price, because they've already been paying those tax rates), or they can lower their margin.
It's true that these taxes make things more expensive for local businesses, but also for all businesses, as they have to employ very expensive tax lawyers to figure out the best places to put their businesses. What a waste of mental effort that could've gone into more productive careers. But that's corporation tax for you.
