If you can get the current quotation of a symbol upon page load from your application server and then the page immediately subscribes to the symbol through Lightstreamer Client library to get the updates, this may be enough in most cases.
There is only the risk of missing an update occurring during this short interval. In this case, until the next update comes, the data on the page would be outdated. In practice, the consequences would be worse for slow symbols, but for them the probability of this to happen would be low.

If this risk is acceptable, then this would keep the Data Adapter code simple (in particular, the Data Adapter could answer false to the isSnapshotAvailable() callback).
Otherwise the Data Adapter, upon each subscribe() invocation, should also take care of obtaining from the feed the current stock price. This may involve some REST call, but the frequency of these calls would be much lower than in a polling strategy: it would be needed only upon page load and, whenever many clients should request the same item at the same time, only for the first one.