Searching for secret doors is a turn-long action, and searching only covers a single 10' section of wall. So generally, no, exploration rate movement does not include searching. But see Holmes and the Commentary.
The DM should only check for a secret door if a player says that the character is searching for one in the correct area. The search takes one turn. Each character has only one chance to find each secret door.
It takes 10 minutes (1 turn) of searching
a 10’ x 10’ area to find a secret door.
Less time will mean automatic failure.
(PRB page 57)
Any character may search for secret doors.
The player must describe the exact area
being searched. This takes about 1 turn for
each 10’ x 10’ area searched. If a player
says “My character will look for secret
doors,” roll 1d6. You should always roll,
whether a secret door is there or not, to
keep the players guessing. If there is a
secret door and the result is a 1, the secret
door has been found. Elves find secret
doors with a roll of 1 or 2. Once a secret
door has been found and its exact location
noted on a map, anyone reading that map
can find the secret door later (no roll
necessary). A secret or hidden door may
not be opened until it has been found.
(DMRB page 16)
D&D Cyclopedia (Alston)
Any character may search for secret doors. The
player must describe the exact area being
searched. This takes about one turn for each
10'x 10' area searched. If a player says "My character
will look for secret doors," roll 1d6. You
should always roll, whether a secret door is there
or not, to keep the players guessing. If there is a
secret door and the result is a 1, the character has
found the secret door. Elves find secret doors
with a roll of 1 or 2.
When PCs want to search for secret doors, they
must tell you what area they are searching. (Each
character can search one 10’ x 10 area per turn.)
Roll 1d6 for each PC searching (even if there are
no secret doors in the area). On a result of 1 (1-2
for elves), the character finds the secret door. The
DM should roll dice even if there is no secret door
to be found, just to keep the players guessing.
(1994 color cover, page 36)
Holmes notes that elves have a 1-2 chance just passing by a secret door; elves searching a 10' section of wall have a 1-4, and everyone else has a 1-2 chance on 1d6. (page 10.) Earlier, it provides a weak guideline of one turn per 10' section of wall. (Page 9.)
Note that Holmes Basic is the outlier - in a Holmes game, elves don't even need to search to have a chance. So, in Holmes, and its OSR derivatives, Elves (and only Elves) get a chance to detect while moving at exploration rates.
In Moldvay, Mentzer, Denning, and Alston, to find secret doors requires active searching, at 1 turn per 10' section of wall.