The Sony Vaio VPCL213FX has an i5-2410M CPU @ 2.3GHZ so your risk to try it. Remember we won't get much if any graphics boost so if you want the speed for gaming, even if you did find an i7 in that class, the game speed won't improve.
If overall speed is the goal, max out the RAM and change the HDD to SSD.
Even that nugget wasn't enough to figure out if the CPU is in a socket. http://ark.intel.com/products/52224/Intel-Core-i5-2410M-Processor-3M-Cache-up-to-2_90-GHz gave 2 packages so which is yours?
If it's the BGA1023, then no. You won't be doing this.
I want to upgrade this all-in-one pc from an i5 to an i7 processor. I have scoured the internet and cannot find a Service Manual or any other information on how to crack the case open and replace/upgrade the processor, if even possible. I will greatly appreciate any assistance/guidance!