As of today, several smartphones (particularly those from Samsung and Sony) ship with the Open Mobile API (as implemented by SEEK-for-Android) for access to at least UICC/SIM based secure elements (some may provide access to other types of secure elements too). The stock ROM of the Galaxy S3, for instance, does contain the Open Mobile API which can be used to access the UICC.
The Open Mobile API is accessible through the package org.simalliance.openmobileapi
. Hence, in order to use the Open Mobile API, you would just need to compile your project against this library (see this explanation). But be careful not to include that library into your APK file, as the implementations on devices often slightly differ from what you get by the SEEK-for-Android project.
Note that alternative/custom ROMs (e.g. CyanogenMod) usually do not include the Open Mobile API, even for those platforms where the stock ROM does. For the S3, you can find a tutorial on how to include the necessary adaptions into CyanogenMod here.
So far this gives you access to the Open Mobile API. However, in order to actually access applications on the UICC, your Android app needs to pass the access control mechanisms of the Open Mobile API. See this explanation. Usually, the stock ROM implementations prefer the access rule file base approach over the ARA applet mechanism. So you have to properly configure those access rules on the UICC/SIM card.