Is it possible to get the arguments of the print function?

Actually, what you're looking for is possible. A simple amount of function environment manipulation and sandboxing should do the job. I've tested some things, and ended up with a pleasing result. Now i don't know how much you know about function environments, so here's a quick review:

Function environments

A function environment is a global table consisting of varying keys and values, that can be accessed while the code is executed. Thanks to some functions given to us by Lua, we can actually change these environments, and quite drastically. To change these environments, and monitor them, we can use setfenv(function, table). The first argument "function", can be a function, or a number. If it's a number, that number represents it's stack level (like getfenv). It's second argument "table", is the environment itself. The setfenv function all together, returns the function given in it's first parameter.

Metatables

Again, I don't know how much you already understand, so here's a quick review on metatables. A metatable is a table consisting of functions (called metamethods) that execute when a table is changed, or referenced in any and every way. Now, because environments are tables, this is a very easy concept to implement in our code.

Server script

-- This can be the function that the client invokes from a remote function.

-- Obviously, the remote name / parent may be different for you. But for demonstration purposes, this is what I'm going with.
local RemoteOutput = game:GetService("ReplicatedStorage").RemoutOutput

-- Err will be our error message, in case something goes wrong
local function GetPrintArgs(CodeSource, Err)
    local Args = {} -- the table containing our print arguments

    -- if the code is faulty, this will be false instead of a function
    if CodeSource then

        -- Now i set the function environment of "CodeSource" (since it's the Lua function our loadstring returned), to a metatable that monitors everything being indexed in that code.

        setfenv(CodeSource, setmetatable({

            -- Set up out index metamethod (if you don't know what this is, use the wiki to search metamethods)

            __index = function(t,k)

                -- If they indexed the environment for print, then...
                if k == "print" then

                    -- Return this function instead of print.
                    return function(...)

                        -- Here i insert in the "Args" table, another table consisting of the arguments the user tried to pass to the print function. Now considering this is all happening in a function, these arguments will be "executed" (i.e, saying 2+2 will return 4, and use of variables is supported)

                        Args[#Args+1] = {...}
                    end
                else

                    -- If the index wasn't print, return this string. According to your request, we just want them to be able to print things, not actually make complex code. This is pretty much a sandbox.

                    return 'Not allowed'
                end
            end
        }))() -- execute the function

    -- If CodeSource isn't a function, then...
    else

        -- Return this error message
        return Err
    end
    return Args -- return the table
end

-- Create the remote function event
Remote.OnServerInvoke = function(Client, Request, Value)

    -- I always make a request variable
    Request = Request:lower()

    -- Simple key to tell us what to do, can be anything.
    if Request == "getprint" then

        -- Remember, loadstring returns 2 values. First one being the function, or false, and the second being the error message, or false.

        local CodeFunc, Err = loadstring(Value)
        return GetPrintArgs(CodeFunc, Err)
    end
end

And now we can make a simple function to "decode" this table, from a local script. Like this:

LocalScript

-- Again, assuming this is the name of the remote function...
local RemoteOutput = game:GetService("ReplicatedStorage").RemoteOutput

local function GetPrintArgs(CodeSource)

    -- Remember, we made "getprint" our key.
    local Args = RemoteOutput:InvokeServer("getprint", CodeSource)

    -- If it's a table, then print everything in it
    if type(Args) == "table" then

        -- Iterate through the table of tables, containing the arguments...
        for i,v in pairs(Args)do
            print(unpack(v)) -- "v" should be a table.
        end
    else
        -- If it's not a table, then print it's single value.
        print(Args)
    end
end

-- Now, let's try it:

local CodeToRun = [[
    local Test = "Hello world"
    print(Test)
    print(workspace)
    print(print)
    print("2+2="..tostring(2+2))
]]

GetPrintArgs(CodeToRun)

---------------
-- Output --
---------------

-- > Hello world
-- > Not allowed
-- > function
-- > 2+2=4

All of this code will run client-sided, as i think you were trying to accomplish. But anyway, hope this helped, I don't know how well i explained this, but i tested it and it works. Let me know if you have any questions.

Would string manipulation suffice for something like this, since you start with the code in string form? It would just involve searching through the code to find "print(" then searching for ')' after, then find the string that appears between them. This seems like an obvious approach to me, so perhaps I don't fully understand. I hope this helps anyway! :D

The simplest solution is to redefine the print function.

For example, you could make it fire a remote event:

function print(...)
    local t = {...}
    for i, v in pairs(t) do
        t[i] = tostring(v)
    end
    game.ReplicatedStorage.Output:FireClient(target player, unpack(t))
end

There is also the LogService which has events that are fired whenever things appear in the output. You could use the normal print and just connect to that, though it might be more difficult to determine which player the output belongs to.