Can you verify partial blockchain information from a third party, without yourself having an entire blockchain?
You could use the Simple Payment Verification, or SPV, protocol.
In a nutshell, this works by synchronizing only the block headers for the longest chain. Once the chain is established, individual transactions are proven by using merkle paths leading up to the merkle root in the previously synced block headers.
Note that this only allows for proof that the supplied data exists in the chain - it does not allow the user to verify that you have provided them with all data - it is possible for operators on SPV systems to Lie by omission and simply claim a transaction or block doesn't exist.
Consumer wallets usually get around that by querying multiple such nodes - if at least one node is honest, the wallet is able to detect the lie.