Why not just use two ADCs (say GPIB from digital multimeters) and use them to collect the data. Afterward, you can just emulate the LIA by empirically determining the phase offset between the signal and reference signal?
The main function of a lock-in amplifier is to determine the amplitude of a narrow band signal buried in wideband noise (SNR's can be -60 dB or lower). It does this my multiplying the input signal by a reference sine wave at the same frequency as the input signal. The output of such a multiplier includes a DC term that is proportional to the amplitude of the input signal. Being DC, it can be low pass filtered as much as desired, the trade off being longer response time. However, using time constants of many seconds provides an effective bandwidth of a fraction of a Hertz. This very low bandwidth effectively filters out most of the input noise which is assumed to be spread over a wide band. Some of the functions of a lock-in amplifier can be done using digital techniques but 2 ADC's are not enough.