Well... it sort of exists already

Note that the Ministry text refers to a
national joint application system - which was just recently introduced -
http://www.yliopistohaku.fi AFAIK it's basically
only for applicants from Finland (even though it does seem to be available in English...)
If you are a foreign applicant, there's the University Admissions application system:
http://www.universityadmissions.fi which, however, is not used by all universities... however IMO it is somewhat unclear whether you can apply or not at
http://www.yliopistohaku.fi as an "international" applicant - if you ask me it's a bit of a mess in that sense. And the search function distantly reminds me of
this nightmarish "search function"

(Just for fun try the "In English" - option... look at the drop-down menus... what a joke

) (edit - just to add that the
koulutusnetti service may be useful as such, if you know exactly what you're looking for, but I'm just wondering why it needs to have that "English" interface - when it really doesn't.)
For international students, the UAS thingy
http://www.admissions.fi seems less complicated as regards when and how to apply... the
national joint application for the UAS's is found at
www.amkhaku.fi, and if you click on "degree programmes" on the front page it takes you to
http://www.admissions.fi.