The short answer is yes. Generally speaking, contributions can be made to an HSA up until the due date of the individual's (employee's) federal income tax return for that particular year. That means for 2018 contributions, individuals can contribute to their HSA until April 15, 2019. Since employers (and other third parties) can contribute to an individual’s HSA on their behalf, that rule includes both employer and employee/individual HSA contributions. So, either the individual or the employer on their behalf can make 2018 contributions up until April 15, 2019. The individual (or the employer on their behalf) should notify the HSA trustee/bank that the contributions relate to 2018. The general idea, though, is that the contributions should be allocated to 2018 (and therefore counted towards the individual’s 2018 HSA contribution limit)—the contributions would not impact the individual’s 2019 HSA contribution limit.
Digging a bit deeper into employer obligations, if the employer’s HSA contributions are running through the cafeteria plan (as are most employer HSA contribution designs), then the employer’s contribution design must not favor highly compensated employees per the Section 125 nondiscrimination rules. Making a 2018 contribution in 2019 for an employee who did not timely establish their HSA would not by itself favor highly compensated employees — it would just be giving the employee the contribution they were otherwise entitled to in 2018. If the employer’s HSA contributions are not running through the cafeteria plan, then the HSA comparability rules apply. Those rules also allow 2019 funding for contributions (plus interest) otherwise due in 2018, but the employer should have sent a notice to the employee at the end of 2018 notifying the employee of their obligation to open the HSA (by the last day of February 2019) before employer HSA contributions can be made. The IRS has a model notice for this notice requirement.
Importantly, regardless of whether the employer’s HSA contributions are run through a cafeteria plan or not, employees may not reimburse themselves from the HSA for medical expenses incurred in 2018 since there was no HSA account set up. This relates back to the general rule on HSA reimbursements (also called “distributions”) — individuals may be reimbursed only for expenses that are incurred after the HSA is established. An HSA is established per state law, so the exact answer might vary. But generally speaking, an HSA is considered established when the employee (or employer on their behalf) completes the proper paperwork or application to create the HSA account and the HSA is funded (once money actually goes into the HSA). So, an employee who failed to take appropriate steps to establish the HSA in 2018 could not be reimbursed through an HSA for medical expenses incurred in 2018. Once the HSA is established and funded in 2019, though, the employee could use HSA reimbursements for any expenses incurred after the 2019 HSA establishment date.