Most card terminal providers accept the usual means of payment such as credit and debit cards. To additionaly accept American Express cards, it is necessary to participate in the American Express partner programme.
In Switzerland, American Express works together with the provider Swisscard AECS GmbH. To become a partner of American Express or to request a consultation, you can contact firstname.lastname@example.org by email.
If intrested, you can find a detailed overview and contact details here: American Express