Correct me if I am wrong but the following answer assumes your wavefunction is a product state between sites (1, 2) and the rest of the system, i.e. the bond dimension of the second link is 1.
Then your state can be expressed as |\psi> = |\phi> |\chi> where |\phi> = α_1|+⟩+α_2∣−⟩+α_3∣p+⟩+α_4∣p-⟩ and |\chi> the rest of the MPS from site 3 onwards.
If you are looking for a_1 you can take the overlap (<+|<\chi|) |\psi> where you will need to construct a two site MPS representing the state |+> and then connect it with a trivial bond dimension 1 bond to the MPS |\chi> such that you form the bra MPS (<+|<\chi|).
The above assumes that the norm of |\chi> is 1 which will be obeyed if the norm of |\phi> and |\psi> are 1.
To connect two MPS along a trivial bond so that you go from an MPS with 2 sites and an MPS with N - 2 sites to an MPS with N sites there is probably an optimal ITensors way to do it but you can create an empty MPS with N sites and fill it in. Maybe this is relevant Tensor product together MPS.
To create a two site MPS representing the state |+> you can either create a two site empty MPS and fill it in by deriving it on paper or you can start from |00> and applying some Hadamard and CNOT gates appropriately.
To get the |\chi> state just take everything from site 3 onwards from the original |\psi> MPS and throw away the trivial left bond on site 3.
In fact I guess if the unitary evolution is indeed maintaining a bond dimension 1 on the even links you can grab the two qubit states (1,2), (3,4) etc from the |\psi> MPS list and perform the overlap with |+> etc.