Can Linear Feedback Shift Register be used to generate pseudo radnom numbers with uniform distribution and zero correlation?
Can Linear Feedback Shift Register be used to generate

pseudo random numbers


Yes,


with uniform distribution


yes,


and zero correlation?


impossible!
The structure of an LFSR says that the next state is just a result of the current state, with a deterministic (even linear) function applied to that.
The state of the LFSR is fully defined by the values in the shift register.
There's a finite number of possible states.
Hence, the LFSR must repeat itself, latest after \$p^N-1\$ numbers generated, where \$N\$ is the length of your shift register, and \$p\$ is the cardinality of the possible values in each shift register (typically, your LFSR is binary, \$p=2\$).
The case where an \$N\$-deep LFSR really only repeats after \$p^N-1\$ values, and not earlier (i.e. isn't caught circling in some sub-field) is a special one, we call the resulting sequence the maximum length sequence.
So, you most definitely have a correlation!
If you want to use LFSRs to generate longer sequences, you can combine two or more; that's not really part of your question, so let me just refer you to Gold Sequences. They have real-world applications as white sequences, used e.g. in spreading systems / CDMA.
Also, if you worry about correlation, you probably also worry about linear dependencies of your values, and then the LFSR simply isn't the right tool; it's, as the name says, an inherently linear thing.