I would assume that intelligence, as we understand it, would be the domain of MIO for Amarr. A bit like the British intelligence services partially developed from their colonial secret police kind organizations. Military-type intelligence (and recon/exploration in non-colonized etc.) belongs to the Imperial Navy. So MIO agents fooling around in W-space is not military ("this system now belongs to the Empire"), but intelligence ops.
For Caldari, the CPF is a counterintelligence expert (MI5 etc.). I'd assume that Ishukone Watch, with its assume high-tech focus, would be pretty good at intelligence gathering, though they don't (didn't) have an intelligence division, only surveillance.
For reference, the following corporations had an intelligence division before the patch, along with their faction, main activities and intelligence division leaders:
Brutor Tribe (Minmatar Political, lead by Kalifljasmur Eindremirek)
CONCORD (CONCORD Political/Military, lead by Kilkkuken Mallen)
Caldari Business Tribunal (Caldari Legal/Bureaucratic, lead by Oksuken Seriya)
DED (CONCORD Military, lead by Banniskore Zabulugi)
Dominations (Angel Military, lead by Tobias Kruzhor)
Federal Intelligence Office (Gallente Military, lead by Essinan Corlobault)
House of Records (Caldari Bureaucratic, lead by Tsurkuma Koruda)
Inner Circle (CONCORD Legal, lead by Tatoh Okkamon)
Jovian Directorate (Jove Political, lead(?) by Nadsa Zimse)
Kaalakiota (Caldari Manufacture/Hi-tech, lead by Paakadan Uotadan)
Krusual Tribe (Minmatar Political, lead by Esbrike Adleine)
Mercantile Club (Caldari Political, lead by Anken Aronainen)
Quafe Company (Gallente Manufacture/Retail, lead by Adrebelle Jaredaille)
Republic Justice Department (Minmatar Legal, lead by Wallekon Kortadaken)
Republic Security Services (Minmatar Military, lead by Bjongausen Agnake)
Sebiestor Tribe (Minmatar Political, lead by Eirulf Vagaldi)
Serpentis Inquest (Serpentis Biotech, lead by Onlar Skadidsar)
Sukuuvestaa Corporation (Caldari Construction/Manufacture, lead by Kasata Kuulemonmaa)
The Scope (Gallente Entertainment, lead by Chalenet Plarrovon)
Theology Council (Amarr Legal, lead by Acol Yarwema)
Vherokior Tribe (Minmatar Political, lead by Oddesulf Keriget)
Factional share (corps with intelligence division / total):
Amarr 1 / 27 (+ 4 Mandate)
Angel 1 / 4
Caldari 5 / 48 (notably KK and SuVee)
CONCORD 3 / 4 (SCC is the CONCORD corp without an intelligence division)
Gallente 3 / 32
Jove 1 / 9 (but possibly the best intelligence division out there)
Minmatar 6 / 21 (note - all original Republican tribes had the intelligence division. Nefantar, Starkmanir and the two Thukker corps not included in number)
Serpentis 1 / 2
Notable missing factions: We know that SOCT and SOE do also do shady stuff. The Syndicate likely as well, and Guristas are also "in the know" of things.