Starting a community wiki question to track down whether at least one country allows this.
TL;DR Easiest to do in Switzerland
Austria
Seems to be possible but requires an in-person visit to a doctor/pharmacy
Belgium
Only possible for Belgian residents
Bulgaria
(no info in English, need a Bulgarian speaker)
Croatia
(no info in English, need a Croatian speaker)
Cyprus
(no info in English, need a Greek speaker)
Czechia
Only available to citizens or local residents
Denmark
Danish residents only
Estonia
(no info in English, need an Estonian speaker)
Finland
Not available yet even for residents
France
Now open to French citizens who got a vaccine outside the EU, no matter where they reside, and to tourists who are already in France or the EU or who plan to arrive soon.
EDIT: France is no longer issuing certificates online.
Germany
Seems to be possible but physical presence is required
if the doctor or pharmacist is presented with vaccination documentation on a vaccination against the SARS-CoV-2 coronavirus and they apply suitable measures to avoid issuing an incorrect COVID-19-Vaccination certificate, in particular to verify the identity of the vaccinated person and the authenticity of the vaccination documentation.
Greece
(no info in English, need a Greek speaker)
Hungary
(no info in English, need a Hungarian speaker)
Iceland
(no info in English, need an Icelandic speaker)
Italy
Italian Citizens vaccinated or recovered from COVID abroad can register with a local health agency in Italy to receive a green pass. This is available even for non-residents, and can be done remotely with some health agencies.
Ireland
Doesn't seem to be possible yet
Latvia
(no info in English, need a Latvian speaker)
Lithuania
Local residents only (requires local sign in to be activated)
Luxembourg
Doesn't seem to be available
Malta
Doesn't seem to issue EU certificates yet?
Netherlands
Citizens/local residents only and requires an in-person visit to Utrecht (see e.g. “Ik ben gevaccineerd in een niet-EU-land. En ik heb daar bewijs van.”)
Norway
Not possible yet
Poland
Only available for Polish vaccines for now
Portugal
Local residents only
Romania
(no info in English, need a Romanian speaker)
Slovakia
Local residents only
Slovenia
(no info in English, need a Slovenian speaker)
Spain
Citizens/local residents only
Sweden
Local vaccines only
Switzerland
For a CHF 30 fee, tourists vaccinated with an EMA-approved vaccine and intending to visit Switzerland can apply for a Swiss COVID certificate online. This certificate is compatible with the EUDCC and should be usable throughout the EU.