Thanks for the reply!  I don't think it's permissions...
The app is running under a normal user, but created specifically for creating the environment for the app (so a non-system, unprivileged, userland user).  Let's call it "[appuser]".
The drivers have been installed by the root user, are owned by root:root but are all 774, so executable by anyone.

sh-4.2$ ls -lAFhZ /opt/cache-odbc/bin/
-rwxrwxr-x root root ?                                cconnect.so*
-rwxrwxr-x root root ?                                cgateiw.so*
-rwxrwxr-x root root ?                                cgate.so*
-rwxrwxr-x root root ?                                cgateur64.so*
-rwxrwxr-x root root ?                                cgateu.so*
-rwxrwxr-x root root ?                                libcachedb.so*
-rwxrwxr-x root root ?                                libcacheodbc35.so*
-rwxrwxr-x root root ?                                libcacheodbciw35.so*
-rwxrwxr-x root root ?                                libcacheodbciw.so*
-rwxrwxr-x root root ?                                libcacheodbc.so*
-rwxrwxr-x root root ?                                libcacheodbcu35.so*
-rwxrwxr-x root root ?                                libcacheodbcur6435.so*
-rwxrwxr-x root root ?                                libcacheodbcur64.so*
-rwxrwxr-x root root ?                                libcacheodbcu.so*
-rwxrwxr-x root root ?                                libcacheodbcuw35.so*
-rwxrwxr-x root root ?                                libcacheodbcuw.so*
-r-xr-xr-x root root ?                                libiodbc.so*
-rwxrwxr-x root root ?                                libodbc.so*

There are no SELinux labels (all "?") , but on this Amazon Linux image, SELinux appears to be disabled (a bit terrifying, but means we can discount it as an issue!)

sh-4.2$ getsebool
getsebool:  SELinux is disabled

If I become the app user (with sudo su - [appuser]) and check the library for dependencies, I get:

[appuser]@ip-172-16-121-87 ~]$ ldd /opt/iris-odbc/bin/irisconnect.so
        linux-vdso.so.1 (0x00007ffc65ceb000)
        libssl.so.10 => /lib64/libssl.so.10 (0x00007f860e377000)
        libcrypto.so.10 => /lib64/libcrypto.so.10 (0x00007f860df20000)
        libdl.so.2 => /lib64/libdl.so.2 (0x00007f860dd1c000)
        librt.so.1 => /lib64/librt.so.1 (0x00007f860db14000)
        libstdc++.so.6 => /lib64/libstdc++.so.6 (0x00007f860d792000)
        libm.so.6 => /lib64/libm.so.6 (0x00007f860d452000)
        libgcc_s.so.1 => /lib64/libgcc_s.so.1 (0x00007f860d23c000)
        libc.so.6 => /lib64/libc.so.6 (0x00007f860ce8f000)
        libgssapi_krb5.so.2 => /lib64/libgssapi_krb5.so.2 (0x00007f860cc43000)
        libkrb5.so.3 => /lib64/libkrb5.so.3 (0x00007f860c95f000)
        libcom_err.so.2 => /lib64/libcom_err.so.2 (0x00007f860c75b000)
        libk5crypto.so.3 => /lib64/libk5crypto.so.3 (0x00007f860c52a000)
        libz.so.1 => /lib64/libz.so.1 (0x00007f860c315000)
        /lib64/ld-linux-x86-64.so.2 (0x00007f860e5e6000)
        libpthread.so.0 => /lib64/libpthread.so.0 (0x00007f860c0f7000)
        libkrb5support.so.0 => /lib64/libkrb5support.so.0 (0x00007f860bee8000)
        libkeyutils.so.1 => /lib64/libkeyutils.so.1 (0x00007f860bce4000)
        libresolv.so.2 => /lib64/libresolv.so.2 (0x00007f860bace000)
        libselinux.so.1 => /lib64/libselinux.so.1 (0x00007f860b8a7000)
        libpcre.so.1 => /lib64/libpcre.so.1 (0x00007f860b643000)

Perhaps strangely, if I ask ldd to check dependencies for the Driver we're actually pointing at from the unixODBC driver manager, it doesn't mention irisconnect.so, but certainly doesn't throw any errors or missing dependencies.

[appuser]@ip-172-16-121-87 ~]$ ldd /opt/iris-odbc/bin/libirisodbcur6435.so
        linux-vdso.so.1 (0x00007ffc455fa000)
        libdl.so.2 => /lib64/libdl.so.2 (0x00007f38debe7000)
        libstdc++.so.6 => /lib64/libstdc++.so.6 (0x00007f38de865000)
        libm.so.6 => /lib64/libm.so.6 (0x00007f38de525000)
        libc.so.6 => /lib64/libc.so.6 (0x00007f38de178000)
        libgcc_s.so.1 => /lib64/libgcc_s.so.1 (0x00007f38ddf62000)
        libpthread.so.0 => /lib64/libpthread.so.0 (0x00007f38ddd44000)
        /lib64/ld-linux-x86-64.so.2 (0x00007f38dedeb000)

So the app user appears to be able read and interrogate the various libraries.

I remain suspicious that, given the lack of an obvious dependency, and given the lack of a path in the trace output, perhaps we simply don't know where to find irisconnect.so, but as mentioned in the earlier post, any attempt to set $LD_LIBRARY_PATH or add a module to ld.so.conf.d seems to make the situation worse, not better.

Thanks for this.  We have tried that and get the following error..  While that documentation appears to be IRIS specific, I fear it is inaccurate...?

ERROR [IM002] [Microsoft][ODBC Driver Manager] Data source name not found and no default driver specified"

The registry lists the following keys for ODBC Drivers, so I presume we're using a registry key lookup (into the 32-bit WOW namespace, given the app is still 32-bit)

[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\ODBC\ODBCINST.INI\ODBC Drivers]
"InterSystems ODBC"="Installed"