Yes you can download the Roku app on your phone or tablet and connect it to your Roku player as long as your Roku and mobile device are connected to the same router. That's a good temporary solution until you find your remote or get a replacement.
To add to what @Emissary35 wrote, there are probably several apps that can act as Roku remotes, but you can find the official one from here:
Roku Mobile App - Free for iOS® or Android™ | Roku
I also have to emphasize what was said about the phone and Roku already being connected to the same network. You can't use the app to initially set up a Roku device or change its Wi-Fi connection if it's not connected, so you may want to pick up a Roku Voice Remote if you can't find yours to have "just in case".
As @Emissary35 said, as long as your Roku is already connected to your home network, you can use the Android or iOS Roku app to control it. If you need to connect your Roku to your net, though, the Roku app won't help.
If you have a Roku with an Ethernet port, connect to that if you can and it should find and connect to your local network without problem the next time you power the Roku up. Then you should be able to use the Roku remote app on your phone again.
Otherwise, you have several additional options, of increasing complexity.
FIRST METHOD
Normally a remote compatible with your Roku model is used to set up the Roku's net connection. This is by far the simplest method.
If necessary, report back with your Roku model (and model number if you have it) and we can advise you on which remote you need.
SECOND METHOD
You will need all of these:
-- A Roku that is still set up for its prior network. If it was no longer connecting to that network when you shut it down, if you have cleared its network settings, or if you have done a factory reset, those settings are gone and your only recourse is to use a compatible Roku remote to set the Roku network connection up.
-- Knowledge of the network name (SSID) and password from your former network that the Roku is currently configured to use. Again, without this, your only recourse is to use a compatible Roku remote to set the Roku network connection up.
-- A phone or other mobile device with the Roku app (not TheRokuChannel app) installed.
-- Access to your new router's control panel. If you don't have this access, scroll down to the third method.
If you have ALL of the above:
Go into your router's control panel and set it to use the same network name (SSID) and password as your previous network.
Pull the power plug on your Roku for a few seconds then plug it back in. When it boots up it should connect to your reconfigured net.
Now when you connect your phone to this same net you should be able to use the Roku app to control your Roku once again. Any other devices that you may have already configured to the new network will need to be set back to the old network connection settings.
THIRD METHOD
(I successfully used this method recently when visiting my timeshare. I brought a Roku from home and needed to configure it for the timeshare's network, but found the remote's batteries were dead, rendering the remote useless.)
This method is more complicated, but if you really need it …
In addition to a Roku still configured for its prior network and knowledge of the prior network's SSID and password as listed for method two, you'll also need:
-- A mobile phone with WiFi hotspot capabilities.
-- A second WiFi-equipped mobile device, either a phone or tablet, with the official Roku (not TheRokuChannel) app installed.
If you have all these things, here's how you do it:
1.) Configure the WiFi hotspot on your mobile phone to use the same network name (SSID) and password from your former home network that the Roku is currently configured to use. If your hotspot can use the 5 GHz WiFi band, switch it to the 2.4 GHz band for compatibility with all Roku models.
2.) After you turn on this hotspot, pull the power plug on your Roku for a few seconds then plug it back in. When the Roku starts up it should connect to the hotspot and through the phone to the internet. You just won't be able to control the Roku yet.
3.) Connect your second device (phone or tablet) to the first phone's WiFi hotspot network.
4.) Now that this second device and your Roku are both connected to the same WiFi hotspot, you can run the Roku app on the second device to gain control of your Roku using the app's remote control capabilities.
5.) Using the app on the second device, set up the new network on the Roku, via Settings > Network > Set up network, using the new network's SSID and password.
6.) When the Roku connects to the new network, disconnect the second mobile device from the WiFi hotspot and connect it to the new home network (SSID and password).
7.) You can now continue to use the Roku app on the second device to operate your Roku on the new network.
8.) The Roku is no longer connecting to your phone's WiFi hotspot, so you can shut that down. If you wish, you now can also connect this phone to your home WiFi and use the Roku app on this phone to control your Roku.
If you use method two or three I'd still suggest acquiring a compatible remote to make it possible to recover if your Roku ever drops its network connection and needs to be reconnected, and much simpler if you need to change to a new network again.