If I understand your situation correctly, your client's site doesn't need to be PCI compliant itself, but your client is responsible for selecting a third party payment service provider (PSP) that is PCI compliant.
The safest way to check for compliance is by looking at the VISA global list of compliant PSPs and confirm that the company you're planning to use is listed.
The list is here https://usa.visa.com/merchants/risk_management/cisp_service_providers.html
It used to be a pdf document that you had to download and search, but they recently switched to an searchable online list.
Your clients acquiring bank (where the funds from card transactions end up) is the final arbiter of what exactly your compliance requirements do or don't include so it's best to check with them too.
Actually I just re-read your original question and your client's site MAY need to be PCI compliant after all, depending on how you're implementing the payment functionality. PCI compliance is required if a site is either PROCESSING, STORING or TRANSMITTING card numbers. From what you say you're not processing or storing card details, but your site may be transmitting them. When your client's customer places an order, if they will type their card number into a form that gets posted to your site and then you resend it to Stripe, then you are now transmitting card details, even if you only have it on your site for a microsecond. However, some PSPs have a way of doing the payment functionality where the card details get posted direct to them and the card details never touch your site. If what Stripe is doing is the latter then my original answer above still applies. if it's the former then your client's site will likely need to have quarterly scans. Again, I go back to the fact that it's the acquiring bank who actually makes the final decision on what your compliance requirements are.